2008
2008
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1.I.1B
2008 commentState de Moivre's Theorem. By evaluating
or otherwise, show that
Hence show that
where is an integer in the range .
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1.I.2A
2008 commentLet be an unitary matrix . Suppose that and are Hermitian matrices such that .
Show that
(i) and commute,
(ii) .
Find and in terms of and , and hence show that and are uniquely determined for a given .
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1.II
2008 comment(a) Use suffix notation to prove that
Hence, or otherwise, expand (i) , (ii) .
(b) Write down the equation of the line that passes through the point a and is parallel to the unit vector .
The lines and in three dimensions pass through and respectively and are parallel to the unit vectors and respectively. Show that a necessary condition for and to intersect is
Why is this condition not sufficient?
In the case in which and are non-parallel and non-intersecting, find an expression for the shortest distance between them.
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1.II.6A
2008 commentA real matrix with elements is said to be upper triangular if whenever . Prove that if and are upper triangular real matrices then so is the matrix product .
Consider the matrix
Show that . Write as a linear combination of and and hence compute explicitly.
For all integers (including negative integers), prove that there exist coefficients and such that
For all integers (including negative integers), show that
Hence derive a set of 3 simultaneous equations for and find their solution.
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1.II
2008 commentProve that any orthonormal vectors in form a basis for .
Let be a real symmetric matrix with orthonormal eigenvectors and corresponding eigenvalues . Obtain coefficients such that
is a solution to the equation
where is a given vector and is a given scalar that is not an eigenvalue of .
How would your answer differ if ?
Find and hence when
in the cases (i) and (ii) .
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1.II.8C
2008 commentProve that the eigenvalues of a Hermitian matrix are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal (i.e. ).
Let be a real non-zero antisymmetric matrix. Show that is Hermitian. Hence show that there exists a (complex) eigenvector such , where is imaginary.
Show further that there exist real vectors and and a real number such that
Show also that has a real eigenvector such that .
Let . By considering the action of on and , show that is a rotation matrix.
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3.I.1E
2008 commentDefine the signature of a permutation , and show that the map is a homomorphism.
Define the alternating group , and prove that it is a subgroup of . Is a normal subgroup of ? Justify your answer.
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3.I.2E
2008 commentWhat is the orthogonal group ? What is the special orthogonal group
Show that every element of the special orthogonal group has an eigenvector with eigenvalue 1 . Is this also true for every element of the orthogonal group ? Justify your answer.
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3.II
2008 commentShow that every Möbius map may be expressed as a composition of maps of the form and (where and are complex numbers).
Which of the following statements are true and which are false? Justify your answers.
(i) Every Möbius map that fixes may be expressed as a composition of maps of the form and (where and are complex numbers).
(ii) Every Möbius map that fixes 0 may be expressed as a composition of maps of the form and (where is a complex number).
(iii) Every Möbius map may be expressed as a composition of maps of the form and (where is a complex number).
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3.II
2008 commentState and prove the orbit-stabilizer theorem. Deduce that if is an element of a finite group then the order of divides the order of
Prove Cauchy's theorem, that if is a prime dividing the order of a finite group then contains an element of order .
For which positive integers does there exist a group of order in which every element (apart from the identity) has order 2?
Give an example of an infinite group in which every element (apart from the identity) has order
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3.II
2008 commentFor a normal subgroup of a group , explain carefully how to make the set of (left) cosets of into a group.
For a subgroup of a group , show that the following are equivalent:
(i) is a normal subgroup of ;
(ii) there exist a group and a homomorphism such that is the kernel of .
Let be a finite group that has a proper subgroup of index (in other words, . Show that if ! then cannot be simple. [Hint: Let act on the set of left cosets of by left multiplication.]
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2008
commentProve that two elements of are conjugate if and only if they have the same cycle type.
Describe (without proof) a necessary and sufficient condition for a permutation to have the same conjugacy class in as it has in .
For which is conjugate (in ) to
For every , show that is conjugate to (in . Exhibit a positive integer and a such that is not conjugate to (in ).
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1.I
2008 commentState the ratio test for the convergence of a series.
Find all real numbers such that the series
converges.
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1.I.4E
2008 commentLet be Riemann integrable, and for set .
Assuming that is continuous, prove that for every the function is differentiable at , with .
If we do not assume that is continuous, must it still be true that is differentiable at every ? Justify your answer.
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1.II
2008 commentInvestigate the convergence of the series (i) (ii)
for positive real values of and .
[You may assume that for any positive real value of for sufficiently large. You may assume standard tests for convergence, provided that they are clearly stated.]
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1.II.10D
2008 comment(a) State and prove the intermediate value theorem.
(b) An interval is a subset of with the property that if and belong to and then also belongs to . Prove that if is an interval and is a continuous function from to then is an interval.
(c) For each of the following three pairs of intervals, either exhibit a continuous function from to such that or explain briefly why no such continuous function exists: (i) ; (ii) ; (iii) .
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1.II.11D
2008 comment(a) Let and be functions from to and suppose that both and are differentiable at the real number . Prove that the product is also differentiable at .
(b) Let be a continuous function from to and let for every . Prove that is differentiable at if and only if either or is differentiable at .
(c) Now let be any continuous function from to and let for every . Prove that is differentiable at if and only if at least one of the following two possibilities occurs:
(i) is differentiable at ;
(ii) and
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1.II.12E
2008 commentLet be a complex power series. Prove that there exists an such that the series converges for every with and diverges for every with .
Find the value of for each of the following power series: (i) ; (ii) .
In each case, determine at which points on the circle the series converges.
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2.I.1A
2008 commentLet be a positive constant. Find the solution to the differential equation
that satisfies and as .
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2.I.2A
2008 commentFind the fixed points of the difference equation
Show that a stable fixed point exists when and also when .
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2.II.5A
2008 commentTwo cups of hot tea at temperatures and cool in a room at ambient constant temperature . Initially .
Cup 1 has cool milk added instantaneously at ; in contrast, cup 2 has cool milk added at a constant rate for . Briefly explain the use of the differential equations
where and are the Dirac delta and Heaviside functions respectively, and is a positive constant.
(i) Show that for
(ii) Determine the jump (discontinuity) condition for at and hence find for .
(iii) Using continuity of at show that for
(iv) Compute for and show that for
(v) Find the time , after , at which .
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2.II.6A
2008 commentThe linear second-order differential equation
has linearly independent solutions and . Define the Wronskian of and .
Suppose that is known. Use the Wronskian to write down a first-order differential equation for . Hence express in terms of and .
Show further that satisfies the differential equation
Verify that is a solution of
Compute the Wronskian and hence determine a second, linearly independent, solution of .
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2.II.7A
2008 commentFind the first three non-zero terms in series solutions and for the differential equation
that satisfy the boundary conditions
where and are constants.
Determine the value of such that the change of variable transforms into a differential equation with constant coefficients. Hence find the general solution of .
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2.II.8A
2008 commentConsider the function
where is a positive constant.
Find the critical points of , assuming . Determine the type of each critical point and sketch contours of constant in the two cases (i) and (ii) .
For describe the subset of the plane on which attains its maximum value.
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4.I.3B
2008 commentTwo particles of masses and have position vectors and at time . The particle of mass experiences a force and the particle of mass experiences a force . Show that the centre of mass moves at a constant velocity, and derive an equation of motion for the relative separation .
Now suppose that , where is a positive constant. The particles are initially at rest a distance apart. Calculate how long it takes before they collide.
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4.I.4B
2008 commentA damped pendulum is described by the equation
where and are real positive constants. Determine the location of all the equilibrium points of the system. Classify the equilibrium points in the two cases and .
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4.II.9B
2008 commentAn octopus of mass swims horizontally in a straight line by jet propulsion. At time the octopus is at rest, and its internal cavity contains a mass of water (so that the mass of the octopus plus water is ). It then starts to move by ejecting the water backwards at a constant rate units of mass per unit time and at a constant speed relative to itself. The speed of the octopus at time is , and the mass of the octopus plus remaining water is . The drag force exerted by the surrounding water on the octopus is , where is a positive constant.
Show that, during ejection of water, the equation of motion is
Once all the water has been ejected, at time , the octopus has attained a velocity . Use dimensional analysis to show that
where and are two dimensionless quantities and is an unknown function. Solve equation (1) to find an explicit expression for , and verify that your answer is of the form given in equation (2).
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4.II.10B
2008 commentA body of mass moves in the gravitational field of a much larger spherical object of mass located at the origin. Starting from the equations of motion
show that:
(i) the body moves in an orbit of the form
where is the constant angular momentum per unit mass, and and are constants;
(ii) the total energy of the body is
A meteorite is moving very far from the Earth with speed , and in the absence of the effect of the Earth's gravitational field would miss the Earth by a shortest distance (measured from the Earth's centre). Show that in the subsequent motion
and
Use equation to find the distance of closest approach, and show that the meteorite will collide with the Earth if
where is the radius of the Earth.
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4.II.11B
2008 commentAn inertial reference frame and another reference frame have a common origin , and rotates with angular velocity with respect to . Show the following:
(i) the rates of change of an arbitrary vector a in frames and are related by
(ii) the accelerations in and are related by
where is the position vector relative to .
A train of mass at latitude in the Northern hemisphere travels North with constant speed along a track which runs North-South. Find the magnitude and direction of the sideways force exerted on the train by the track.
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4.II.12B
2008 commentA uniform solid sphere has mass and radius . Calculate the moment of inertia of the sphere about an axis through its centre.
A long hollow circular cylinder of radius (where ) is held fixed with its axis horizontal. The sphere is held initially at rest in contact with the inner surface of the cylinder at , where and is the angle between the line joining the centre of the sphere to the cylinder axis and the downward vertical, as shown in the figure.

The sphere is then released, and rolls without slipping. Show that the angular velocity of the sphere is
Show further that the time, , it takes the sphere to reach is
If, instead, the cylinder and sphere surfaces are highly polished, so that the sphere now slides without rolling, find the time, , it takes to reach .
Without further calculation, explain qualitatively how your answers for and would be affected if the solid sphere were replaced by a hollow spherical shell of the same radius and mass.
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4.I.1D
2008 commentLet and be non-empty sets and let and be two functions. For each of the following statements, give either a brief justification or a counterexample.
(i) If is an injection and is a surjection, then is a surjection.
(ii) If is an injection and is an injection, then there exists a function such that is equal to the identity function on .
(iii) If and are subsets of then .
(iv) If and are subsets of then .
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4.I.2D
2008 comment(a) Let be an equivalence relation on a set . What is an equivalence class of Prove that the equivalence classes of form a partition of .
(b) Let be the set of all positive integers. Let a relation be defined on by setting if and only if for some (not necessarily positive) integer . Prove that is an equivalence relation, and give an example of a set that contains precisely one element of each equivalence class.
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4.II.5D
2008 comment(a) Define the notion of a countable set, and prove that the set is countable. Deduce that if and are countable sets then is countable, and also that a countable union of countable sets is countable.
(b) If is any set of real numbers, define to be the set of all real roots of non-zero polynomials that have coefficients in . Now suppose that is a countable set of real numbers and define a sequence by letting each be equal to . Prove that the union is countable.
(c) Deduce that there is a countable set that contains the real numbers 1 and and has the further property that if is any non-zero polynomial with coefficients in , then all real roots of belong to .
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4.II.6D
2008 comment(a) Let and be integers with and let be their highest common factor. For any integer , prove that is a multiple of if and only if there exists an integer satisfying the equation exactly solutions to the equation that are distinct .
Deduce that the equation has a solution if and only if .
(b) Let be a prime and let be the multiplicative group of non-zero integers . An element of is called a th power if for some integer . It can be shown that has a generator: that is, an element such that every element of is a power of . Assuming this result, deduce that an element of is a th power if and only if , where is now the highest common factor of and .
(c) How many 437th powers are there mod 1013? [You may assume that 1013 is a prime number.]
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4.II.7D
2008 comment(a) Let be a field such that the equation has no solution in . Prove that if and are elements of such that , then both and must equal 0 .
Prove that can be made into a field, with operations
and
(b) Let be a prime of the form . Prove that is not a square , and deduce that there exists a field with exactly elements.
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4.II.8D
2008 commentLet be a positive integer. For every positive integer , define a number by the formula
Prove by induction that
for every , and hence evaluate the infinite .
Let be a sequence of integers satisfying the inequality for every . Prove that the series ! is convergent. Prove also that its limit is irrational if and only if for infinitely many and for infinitely many .
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2.I.3F
2008 commentThere are socks in a drawer, three of which are red and the rest black. John chooses his socks by selecting two at random from the drawer and puts them on. He is three times more likely to wear socks of different colours than to wear matching red socks. Find .
For this value of , what is the probability that John wears matching black socks?
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2.I.4F
2008 commentA standard six-sided die is thrown. Calculate the mean and variance of the number shown.
The die is thrown times. By using Chebyshev's inequality, find an such that
where is the total of the numbers shown over the throws.
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2.II.9F
2008 commentA population evolves in generations. Let be the number of members in the th generation, with . Each member of the th generation gives birth to a family, possibly empty, of members of the th generation; the size of this family is a random variable and we assume that the family sizes of all individuals form a collection of independent identically distributed random variables each with generating function .
Let be the generating function of . State and prove a formula for in terms of . Determine the mean of in terms of the mean of .
Suppose that has a Poisson distribution with mean . Find an expression for in terms of , where is the probability that the population becomes extinct by the th generation.
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2.II.10F
2008 commentand play a series of games. The games are independent, and each is won by with probability and by with probability . The players stop when the number of wins by one player is three greater than the number of wins by the other player. The player with the greater number of wins is then declared overall winner.
(i) Find the probability that exactly 5 games are played.
(ii) Find the probability that is the overall winner.
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2.II.11F
2008 commentLet and have the bivariate normal density function
for fixed . Let . Show that and are independent variables. Hence, or otherwise, determine
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2.II.12F
2008 commentThe discrete random variable has distribution given by
where . Determine the mean and variance of .
A fair die is rolled until all 6 scores have occurred. Find the mean and standard deviation of the number of rolls required.
[Hint:
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3.I.3C
2008 commentA curve is given in terms of a parameter by
(i) Find the arc length of the curve between the points with and .
(ii) Find the unit tangent vector at the point with parameter , and show that the principal normal is orthogonal to the direction at each point on the curve.
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3.I.4C
2008 commentWhat does it mean to say that transforms as a second rank tensor?
If transforms as a second rank tensor, show that transforms as a vector.
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3.II.9C
2008 commentLet , where is the position vector and is a uniform vector field.
(i) Use the divergence theorem to evaluate the surface integral , where is the closed surface of the cube with vertices .
(ii) Show that . Show further that the scalar field given by
satisfies . Describe geometrically the surfaces of constant .
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3.II.10C
2008 commentFind the effect of a rotation by about the -axis on the tensor
Hence show that the most general isotropic tensor of rank 2 is , where is an arbitrary scalar.
Prove that there is no non-zero isotropic vector, and write down without proof the most general isotropic tensor of rank 3 .
Deduce that if is an isotropic tensor then the following results hold, for some scalars and : (i) ; (ii) ; (iii) .
Verify these three results in the case , expressing and in terms of and .
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3.II.11C
2008 commentLet be a volume in bounded by a closed surface .
(a) Let and be twice differentiable scalar fields such that on and in . Show that
(b) Let be the sphere . Evaluate the integral
in the cases where and are given in spherical polar coordinates by: (i) ; (ii) ; (iii) .
Comment on your results in the light of part (a).
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3.II.12C
2008 commentLet be the closed planar region given by
(i) Evaluate by means of a suitable change of variables the integral
(ii) Let be the boundary of . Evaluate the line integral
by integrating along each section of the boundary.
(iii) Comment on your results.
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1.I.1E
2008 commentLet be an matrix over . What does it mean to say that is an eigenvalue of ? Show that has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.
(i) If is Hermitian, all eigenvalues of are real.
(ii) If all eigenvalues of are real, is Hermitian.
(iii) If all entries of are real and positive, all eigenvalues of have positive real part.
(iv) If and have the same trace and determinant then they have the same eigenvalues.
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1.II.9E
2008 commentLet be an matrix of real numbers. Define the row rank and column rank of and show that they are equal.
Show that if a matrix is obtained from by elementary row and column operations then .
Let and be matrices. Show that the matrices and have the same rank.
Hence, or otherwise, prove that
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2.I.1E
2008 commentSuppose that and are finite-dimensional vector spaces over . What does it mean to say that is a linear map? State the rank-nullity formula. Using it, or otherwise, prove that a linear map is surjective if, and only if, it is injective.
Suppose that is a linear map which has a right inverse, that is to say there is a linear map such that , the identity map. Show that .
Suppose that and are two matrices over such that . Prove that .
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2.II.10E
2008 commentDefine the determinant of an square matrix over the complex numbers. If and are two such matrices, show that .
Write for the characteristic polynomial of a matrix . Let be matrices and suppose that is nonsingular. Show that . Taking for appropriate values of , or otherwise, deduce that .
Show that if then . Which of the following statements is true for all matrices ? Justify your answers.
(i) ;
(ii) .
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3.II.10E
2008 commentLet or . What is meant by a quadratic form ? Show that there is a basis for such that, writing , we have for some scalars
Suppose that . Define the rank and signature of and compute these quantities for the form given by .
Suppose now that and that are quadratic forms. If , show that there is some nonzero such that .
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4.I.1E
2008 commentDescribe (without proof) what it means to put an matrix of complex numbers into Jordan normal form. Explain (without proof) the sense in which the Jordan normal form is unique.
Put the following matrix in Jordan normal form:
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4.II.10E
2008 commentWhat is meant by a Hermitian matrix? Show that if is Hermitian then all its eigenvalues are real and that there is an orthonormal basis for consisting of eigenvectors of .
A Hermitian matrix is said to be positive definite if for all . We write in this case. Show that is positive definite if, and only if, all of its eigenvalues are positive. Show that if then has a unique positive definite square root .
Let be two positive definite Hermitian matrices with . Writing and , show that . By considering eigenvalues of , or otherwise, show that .
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1.II.10G
2008 comment(i) Show that is not simple.
(ii) Show that the group Rot of rotational symmetries of a regular dodecahedron is a simple group of order 60 .
(iii) Show that is isomorphic to .
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2.I.2G
2008 commentWhat does it means to say that a complex number is algebraic over ? Define the minimal polynomial of .
Suppose that satisfies a nonconstant polynomial which is irreducible over . Show that there is an isomorphism .
[You may assume standard results about unique factorisation, including Gauss's lemma.]
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3.I.1G
2008 commentLet be the abelian group generated by elements subject to the relations
Express as a product of cyclic groups, and find the number of elements of of order 2 .
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2008
commentWhat is a Euclidean domain? Show that a Euclidean domain is a principal ideal domain.
Show that is not a Euclidean domain (for any choice of norm), but that the ring
is Euclidean for the norm function .
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2.II.11G
2008 commentLet be a field. Prove that every ideal of the ring is finitely generated.
Consider the set
Show that is a subring of which is not Noetherian.
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4.I.2G
2008 commentLet be an integer. Show that the polynomial is irreducible over if and only if is prime.
[You may use Eisenstein's criterion without proof.]
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4.II.11G
2008 commentLet be a ring and an -module. What does it mean to say that is a free -module? Show that is free if there exists a submodule such that both and are free.
Let and be -modules, and submodules. Suppose that and . Determine (by proof or counterexample) which of the following statements holds:
(1) If is free then .
(2) If is free then .
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1.I.2G
2008 commentShow that any element of is a rotation, and that it can be written as the product of two reflections.
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2.II.12G
2008 commentShow that the area of a spherical triangle with angles is . Hence derive the formula for the area of a convex spherical -gon.
Deduce Euler's formula for a decomposition of a sphere into convex polygons with a total of edges and vertices.
A sphere is decomposed into convex polygons, comprising quadrilaterals, pentagons and hexagons, in such a way that at each vertex precisely three edges meet. Show that there are at most 7 possibilities for the pair , and that at least 3 of these do occur.
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3.I.2G
2008 commentA smooth surface in has parametrization
Show that a unit normal vector at the point is
and that the curvature is .
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3.II.12G
2008 commentLet be the unit disc model of the hyperbolic plane, with metric
(i) Show that the group of Möbius transformations mapping to itself is the group of transformations
where and .
(ii) Assuming that the transformations in (i) are isometries of , show that any hyperbolic circle in is a Euclidean circle.
(iii) Let and be points on the unit circle with . Show that the hyperbolic distance from to the hyperbolic line is given by
(iv) Deduce that if then no hyperbolic open disc of radius is contained in a hyperbolic triangle.
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4.II.12G
2008 commentLet be a curve on a smoothly embedded surface . Define the energy of . Show that if is a stationary point for the energy for proper variations of , then satisfies the geodesic equations
where in terms of a smooth parametrization for , with first fundamental form .
Now suppose that for every the curves are geodesics.
(i) Show that and .
(ii) Suppose moreover that the angle between the curves is independent of and . Show that .
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1.II.11F
2008 commentState and prove the Contraction Mapping Theorem.
Let be a nonempty complete metric space and a mapping such that, for some , the th iterate of (that is, composed with itself times) is a contraction mapping. Show that has a unique fixed point.
Now let be the space of all continuous real-valued functions on , equipped with the uniform norm , and let be a continuous function satisfying the Lipschitz condition
for all and all , where is a constant. Let be defined by
where is a fixed continuous function on . Show by induction on that
for all and all . Deduce that the integral equation
has a unique continuous solution on .
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2.I.3F
2008 commentExplain what is meant by the statement that a sequence of functions defined on an interval converges uniformly to a function . If converges uniformly to , and each is continuous on , prove that is continuous on .
Now suppose additionally that is a sequence of points of converging to a limit . Prove that .
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2.II.13F
2008 commentLet be a sequence of real-valued functions defined on a subset of . Suppose that for all and all we have , where converges. Prove that converges uniformly on .
Now let , and consider the series , where and
for . Show that the series converges uniformly on for any real number . Deduce that is a continuous function on . Does the series converge uniformly on ? Justify your answer.
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2008
commentExplain what it means for a function of two variables to be differentiable at a point . If is differentiable at , show that for any the function defined by
is differentiable at , and find its derivative in terms of the partial derivatives of at .
Consider the function defined by

Is differentiable at ? Justify your answer.
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3.II.13F
2008 commentLet be a function, and a point of . Prove that if the partial derivatives of exist in some open disc around and are continuous at , then is differentiable at .
Now let denote the vector space of all real matrices, and let be the function assigning to each matrix its determinant. Show that is differentiable at the identity matrix , and that is the linear map . Deduce that is differentiable at any invertible matrix , and that is the linear map
Show also that if is a matrix with , then is invertible. Deduce that is twice differentiable at , and find as a bilinear map .
[You may assume that the norm on is complete, and that it satisfies the inequality for any two matrices and
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4.I.3F
2008 commentLet be the vector space of all continuous real-valued functions on the unit interval . Show that the functions
both define norms on .
Consider the sequence defined by . Does converge in the norm ? Does it converge in the norm ? Justify your answers.
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4.II.13F
2008 commentExplain what it means for two norms on a real vector space to be Lipschitz equivalent. Show that if two norms are Lipschitz equivalent, then one is complete if and only if the other is.
Let be an arbitrary norm on the finite-dimensional space , and let denote the standard (Euclidean) norm. Show that for every with , we have
where is the standard basis for , and deduce that the function is continuous with respect to . Hence show that there exists a constant such that for all with , and deduce that and are Lipschitz equivalent.
[You may assume the Bolzano-Weierstrass Theorem.]
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1.II.12F
2008 commentWrite down the definition of a topology on a set .
For each of the following families of subsets of , determine whether is a topology on . In the cases where the answer is 'yes', determine also whether is a Hausdorff space and whether it is compact.
(a) : either is finite or .
(b) : either is finite or .
(c) : there exists such that, for all .
(d) : for all , there exists such that .
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2.I.4F
2008 commentStating carefully any results on compactness which you use, show that if is a compact space, is a Hausdorff space and is bijective and continuous, then is a homeomorphism.
Hence or otherwise show that the unit circle is homeomorphic to the quotient space , where is the equivalence relation defined by
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2008
commentExplain what it means for a topological space to be connected.
Are the following subspaces of the unit square connected? Justify your answers.
(a) , and .
(b) or and .
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4.II.14F
2008 commentExplain what is meant by a base for a topology. Illustrate your definition by describing bases for the topology induced by a metric on a set, and for the product topology on the cartesian product of two topological spaces.
A topological space is said to be separable if there is a countable subset which is dense, i.e. such that for every nonempty . Show that a product of two separable spaces is separable. Show also that a metric space is separable if and only if its topology has a countable base, and deduce that every subspace of a separable metric space is separable.
Now let with the topology having as a base the set of all half-open intervals
with . Show that is separable, but that the subspace of is not separable.
[You may assume standard results on countability.]
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1.I.3C
2008 commentGiven that is an analytic function, show that the mapping
(a) preserves angles between smooth curves intersecting at if ;
(b) has Jacobian given by .
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1.II.13C
2008 commentBy a suitable choice of contour show the following:
(a)
where ,
(b)
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2.II.14C
2008 commentLet . Find the first three terms in the Laurent expansion for valid for .
Now let be a positive integer, and define
Show that the singularities of in are all removable. By expanding as a Laurent series valid for , and as a Taylor series valid for , find the coefficients of for in the Laurent series for valid for .
By estimating an appropriate integral around the contour , show that
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3.II.14E
2008 commentState and prove Rouché's theorem, and use it to count the number of zeros of inside the annulus .
Let be a sequence of polynomials of degree at most with the property that converges uniformly on compact subsets of as . Prove that there is a polynomial of degree at most such that uniformly on compact subsets of . [If you use any results about uniform convergence of analytic functions, you should prove them.]
Suppose that has distinct roots . Using Rouché's theorem, or otherwise, show that for each there is a sequence such that and as .
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4.I.4E
2008 commentSuppose that and are two functions which are analytic on the whole complex plane . Suppose that there is a sequence of distinct points with such that . Show that for all . [You may assume any results on Taylor expansions you need, provided they are clearly stated.]
What happens if the assumption that is dropped?
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3.I.5C
2008 commentUsing the contour integration formula for the inversion of Laplace transforms find the inverse Laplace transforms of the following functions: (a) real and non-zero , (b) .
[You may use the fact that .]
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4.II.15C
2008 commentLet be the domain (i.e., cut along the negative -axis). Show, by a suitable choice of branch, that the mapping
maps onto the strip .
How would a different choice of branch change the result?
Let be the domain . Find an analytic transformation that maps to , where is the strip defined above.
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1.II.14D
2008 commentWrite down the Euler-Lagrange equation for the variational problem for that extremizes the integral defined as
with boundary conditions , where and are positive constants such that , with . Find a first integral of the equation when is independent of , i.e. .
A light ray moves in the plane from to with speed taking a time . Show that the equation of the path that makes an extremum satisfies
where is a constant and write down an integral relating and .
When where is a constant and , show that the path is given by
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2008
commentDescribe briefly the method of Lagrange multipliers for finding the stationary values of a function subject to a constraint .
Use the method to find the largest possible volume of a circular cylinder that has surface area (including both ends).
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2.II.15D
2008 comment(a) Legendre's equation may be written in the form
Show that there is a series solution for of the form
where the satisfy the recurrence relation
Hence deduce that there are solutions for that are polynomials of degree , provided that . Given that is then chosen so that , find the explicit form for .
(b) Laplace's equation for in spherical polar coordinates may be written in the axisymmetric case as
where .
Write down without proof the general form of the solution obtained by the method of separation of variables. Use it to find the form of exterior to the sphere that satisfies the boundary conditions, , and .
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3.I.6D
2008 commentLet be the operator
on functions satisfying and .
Given that the Green's function for satisfies
show that a solution of
for a given function , is given by
Indicate why this solution is unique.
Show further that the Green's function is given by
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3.II.15D
2008 commentLet and be the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville system
where
with and . The boundary conditions on are that .
Show that two distinct eigenfunctions are orthogonal in the sense that
Show also that if has the form
with being independent of , then
Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation,
that satisfies the boundary conditions given above is such that
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4.I.5A
2008 commentFind the half-range Fourier cosine series for . Hence show that
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4.II.16A
2008 commentAssume satisfies
and that the series
converges uniformly in .
If is the Fourier transform of , prove that
[Hint: prove that is periodic and express its Fourier expansion coefficients in terms of .
In the case that , evaluate the sum
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1.II.15A
2008 commentThe radial wavefunction for the hydrogen atom satisfies the equation
With reference to the general form for the time-independent Schrödinger equation, explain the origin of each term. What are the allowed values of ?
The lowest-energy bound-state solution of , for given , has the form . Find and and the corresponding energy in terms of .
A hydrogen atom makes a transition between two such states corresponding to and . What is the frequency of the emitted photon?
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2.II.16A
2008 commentGive the physical interpretation of the expression
for an observable , where is a Hermitian operator and is normalised. By considering the norm of the state for two observables and , and real values of , show that
Deduce the uncertainty relation
where is the uncertainty of .
A particle of mass moves in one dimension under the influence of potential . By considering the commutator , show that the expectation value of the Hamiltonian satisfies
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3.I.7A
2008 commentWrite down a formula for the orbital angular momentum operator . Show that its components satisfy
If , show that are also eigenvectors of , and find their eigenvalues.
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3.II.16A
2008 commentWhat is the probability current for a particle of mass , wavefunction , moving in one dimension?
A particle of energy is incident from on a barrier given by
where . What are the conditions satisfied by at and ? Write down the form taken by the wavefunction in the regions and distinguishing between the cases and . For both cases, use your expressions for to calculate the probability currents in these two regions.
Define the reflection and transmission coefficients, and . Using current conservation, show that the expressions you have derived satisfy . Show that if .
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4.I.6A
2008 commentWhat is meant by a stationary state? What form does the wavefunction take in such a state? A particle has wavefunction , such that
where and are normalised eigenstates of the Hamiltonian with energies and . Write down at time . Show that the expectation value of at time is
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1.II.16B
2008 commentSuppose that the current density is constant in time but the charge density is not.
(i) Show that is a linear function of time:
where is the time derivative of at time .
(ii) The magnetic induction due to a current density can be written as
Show that this can also be written as
(iii) Assuming that vanishes at infinity, show that the curl of the magnetic field in (1) can be written as
[You may find useful the identities and also
(iv) Show that the second term on the right hand side of (2) can be expressed in terms of the time derivative of the electric field in such a way that itself obeys Ampère's law with Maxwell's displacement current term, i.e. .
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2.I.6B
2008 commentGiven the electric potential of a dipole
where is the magnitude of the dipole moment, calculate the corresponding electric field and show that it can be written as
where is the unit vector in the radial direction.
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2.II.17B
2008 commentTwo perfectly conducting rails are placed on the -plane, one coincident with the -axis, starting at , the other parallel to the first rail a distance apart, starting at . A resistor is connected across the rails between and , and a uniform magnetic field , where is the unit vector along the -axis and , fills the entire region of space. A metal bar of negligible resistance and mass slides without friction on the two rails, lying perpendicular to both of them in such a way that it closes the circuit formed by the rails and the resistor. The bar moves with speed to the right such that the area of the loop becomes larger with time.
(i) Calculate the current in the resistor and indicate its direction of flow in a diagram of the system.
(ii) Show that the magnetic force on the bar is
(iii) Assume that the bar starts moving with initial speed at time , and is then left to slide freely. Using your result from part (ii) and Newton's laws show that its velocity at the time is
(iv) By calculating the total energy delivered to the resistor, verify that energy is conserved.
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3.II.17B
2008 comment(i) From Maxwell's equations in vacuum,
obtain the wave equation for the electric field E. [You may find the following identity useful:
(ii) If the electric and magnetic fields of a monochromatic plane wave in vacuum are
show that the corresponding electromagnetic waves are transverse (that is, both fields have no component in the direction of propagation).
(iii) Use Faraday's law for these fields to show that
(iv) Explain with symmetry arguments how these results generalise to
where is the polarisation vector, i.e., the unit vector perpendicular to the direction of motion and along the direction of the electric field, and is the unit vector in the direction of propagation of the wave.
(v) Using Maxwell's equations in vacuum prove that:
where is the closed volume and is the bounding surface. Comment on the differing time dependencies of the left-hand-side of (1) for the case of (a) linearly-polarized and (b) circularly-polarized monochromatic plane waves.
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4.I.7B
2008 commentThe energy stored in a static electric field is
where is the associated electric potential, , and is the volume charge density.
(i) Assuming that the energy is calculated over all space and that vanishes at infinity, show that the energy can be written as
(ii) Find the electric field produced by a spherical shell with total charge and radius , assuming it to vanish inside the shell. Find the energy stored in the electric field.
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1.I.4C
2008 commentIn an inertial frame a photon of energy is observed to travel at an angle relative to the -axis. The inertial frame moves relative to at velocity in the direction and the -axis of is taken parallel to the -axis of . Observed in , the photon has energy and travels at an angle relative to the -axis. Show that
where .
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2.I.7C
2008 commentA photon of energy collides with a particle of rest mass , which is at rest. The final state consists of a photon and a particle of rest mass . Show that the minimum value of for which it is possible for this reaction to take place is
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4.II.17C
2008 commentWrite down the formulae for the one-dimensional Lorentz transformation for frames moving with relative velocity along the -direction. Derive the relativistic formula for the addition of velocities and .
A train, of proper length , travels past a station at velocity . The origin of the inertial frame , with coordinates , in which the train is stationary, is located at the mid-point of the train. The origin of the inertial frame , with coordinates , in which the station is stationary, is located at the mid-point of the platform. Coordinates are chosen such that when the origins coincide then .
Observers A and B, stationary in , are located, respectively, at the front and rear of the train. Observer C, stationary in , is located at the origin of . At , C sends two signals, which both travel at speed , where , one directed towards and the other towards , who receive the signals at respective times and . observes these events to occur, respectively, at times and . At also observes that the two ends of the platform coincide with the positions of and .
(a) Draw two space-time diagrams, one for and the other for , showing the trajectories of the observers and the events that take place.
(b) What is the length of the platform in terms of ? Briefly illustrate your answer by reference to the space-time diagrams.
(c) Calculate the time differences and .
(d) Setting , use this example to discuss briefly the fact that two events observed to be simultaneous in one frame need not be observed to be simultaneous in another.
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1.I.5B
2008 commentVerify that the two-dimensional flow given in Cartesian coordinates by
satisfies . Find the stream function . Sketch the streamlines.
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1.II.17B
2008 commentTwo incompressible fluids flow in infinite horizontal streams, the plane of contact being , with positive upwards. The flow is given by
where is the unit vector in the positive direction. The upper fluid has density and pressure , the lower has density and pressure , where is a constant and is the acceleration due to gravity.
(i) Consider a perturbation to the flat surface of the form
State the kinematic boundary conditions on the velocity potentials that hold on the interface in the two domains, and show by linearising in that they reduce to
(ii) State the dynamic boundary condition on the perturbed interface, and show by linearising in that it reduces to
(iii) Use the velocity potentials
where , and the conditions in (i) and (ii) to perform a stability analysis. Show that the relation between and is
Find the criterion for instability.
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2.I.8B
2008 comment(i) Show that for a two-dimensional incompressible flow , the vorticity is given by where is the stream function.
(ii) Express the -component of the vorticity equation
in terms of the stream function .
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3.II.18B
2008 commentAn ideal liquid contained within a closed circular cylinder of radius rotates about the axis of the cylinder (assume this axis to be in the vertical -direction).
(i) Prove that the equation of continuity and the boundary conditions are satisfied by the velocity , where is the angular velocity, with the unit vector in the -direction, which depends only on time, and is the position vector measured from a point on the axis of rotation.
(ii) Calculate the angular momentum per unit length of the cylinder.
(iii) Suppose the the liquid starts from rest and flows under the action of an external force per unit mass . By taking the curl of the Euler equation, prove that
(iv) Find the pressure.
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4.II.18B
2008 comment(i) Starting from Euler's equation for an incompressible fluid show that for potential flow with ,
where , the body force per unit mass is and is an arbitrary function of time.
(ii) Hence show that, for the steady flow of a liquid of density through a pipe of varying cross-section that is subject to a pressure difference between its two ends, the mass flow through the pipe per unit time is given by
where and are the cross-sectional areas of the two ends.
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1.I.6D
2008 commentShow that if , where is a lower triangular matrix with all elements on the main diagonal being unity and is a diagonal matrix with positive elements, then is positive definite. Find and the corresponding when
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2.II.18D
2008 comment(a) A Householder transformation (reflection) is given by
where , and is the unit matrix and is a non-zero vector which has norm . Show that is orthogonal.
(b) Suppose that and with . Show that if minimises then it also minimises , where is an arbitrary orthogonal matrix.
(c) Using Householder reflection, find the that minimises when
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3.II.19D
2008 commentStarting from the Taylor formula for with an integral remainder term, show that the error of an approximant can be written in the form (Peano kernel theorem)
when , which is identically zero if is a polynomial of degree , satisfies conditions that you should specify. Give an expression for .
Hence determine the minimum value of in the inequality
when
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4.I.8D
2008 commentShow that the Chebyshev polynomials, obey the orthogonality relation
State briefly how an optimal choice of the parameters is made in the Gaussian quadrature formula
Find these parameters for the case .
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2008
commentA Bayesian statistician observes a random sample drawn from a distribution. He has a prior density for the unknown parameters of the form
where and are constants which he chooses. Show that after observing his posterior density is again of the form
where you should find explicitly the form of and .
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1.II.18H
2008 commentSuppose that is a sample of size with common distribution, and is an independent sample of size from a distribution.
(i) Find (with careful justification) the form of the size- likelihood-ratio test of the null hypothesis against alternative unrestricted.
(ii) Find the form of the size- likelihood-ratio test of the hypothesis
against unrestricted, where is a given constant.
Compare the critical regions you obtain in (i) and (ii) and comment briefly.
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2.II.19H
2008 commentSuppose that the joint distribution of random variables taking values in is given by the joint probability generating function
where the unknown parameters and are positive, and satisfy the inequality . Find . Prove that the probability mass function of is
and prove that the maximum-likelihood estimators of and based on a sample of size drawn from the distribution are
where (respectively, ) is the sample mean of (respectively, ).
By considering or otherwise, prove that the maximum-likelihood estimator is biased. Stating clearly any results to which you appeal, prove that as , making clear the sense in which this convergence happens.
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2008
commentIf is a sample from a density with unknown, what is a confidence set for ?
In the case where the are independent random variables with known, unknown, find (in terms of ) how large the size of the sample must be in order for there to exist a confidence interval for of length no more than some given .
[Hint: If then
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4.II.19H
2008 comment(i) Consider the linear model
where observations , depend on known explanatory variables , , and independent random variables .
Derive the maximum-likelihood estimators of and .
Stating clearly any results you require about the distribution of the maximum-likelihood estimators of and , explain how to construct a test of the hypothesis that against an unrestricted alternative.
(ii) A simple ballistic theory predicts that the range of a gun fired at angle of elevation should be given by the formula
where is the muzzle velocity, and is the gravitational acceleration. Shells are fired at 9 different elevations, and the ranges observed are as follows:
The model
is proposed. Using the theory of part (i) above, find expressions for the maximumlikelihood estimators of and .
The -test of the null hypothesis that against an unrestricted alternative does not reject the null hypothesis. Would you be willing to accept the model ? Briefly explain your answer.
[You may need the following summary statistics of the data. If , then 17186. ]
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1.I.8H
2008 commentState the Lagrangian Sufficiency Theorem for the maximization over of subject to the constraint .
For each , solve
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2.I.9H
2008 commentGoods from three warehouses have to be delivered to five shops, the cost of transporting one unit of good from warehouse to shop being , where
The requirements of the five shops are respectively and 10 units of the good, and each warehouse holds a stock of 15 units. Find a minimal-cost allocation of goods from warehouses to shops and its associated cost.
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3.II.20H
2008 commentUse the simplex algorithm to solve the problem
subject to , and
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4.II.20H
2008 comment(i) Suppose that , and are continuously differentiable. Suppose that the problem
subject to
is solved by a unique for each , and that there exists a unique such that
Assuming that and are continuously differentiable, prove that
(ii) The output of a firm is a function of the capital deployed, and the amount of labour employed, given by
where . The firm's manager has to optimize the output subject to the budget constraint
where is the wage rate and is the available budget. By casting the problem in Lagrangian form, find the optimal solution and verify the relation .
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1.II.19H
2008 commentThe village green is ringed by a fence with fenceposts, labelled . The village idiot is given a pot of paint and a brush, and started at post 0 with instructions to paint all the posts. He paints post 0 , and then chooses one of the two nearest neighbours, 1 or , with equal probability, moving to the chosen post and painting it. After painting a post, he chooses with equal probability one of the two nearest neighbours, moves there and paints it (regardless of whether it is already painted). Find the distribution of the last post unpainted.
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2.II.20H
2008 commentA Markov chain with state-space has non-zero transition probabilities and
Prove that this chain is recurrent if and only if
Prove that this chain is positive-recurrent if and only if
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3.I.9H
2008 commentWhat does it mean to say that a Markov chain is recurrent?
Stating clearly any general results to which you appeal, prove that the symmetric simple random walk on is recurrent.
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4.I.9H
2008 commentA Markov chain on the state-space has transition matrix
Classify the chain into its communicating classes, deciding for each what the period is, and whether the class is recurrent.
For each say whether the exists, and evaluate the limit when it does exist.
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1.I.1H
2008 commentDefine the continued fraction of a real number .
Compute the continued fraction of .
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2.I.1H
2008 commentWhat does it mean for a positive definite quadratic form with integer coefficients to be reduced?
Show that there are precisely three reduced forms of this type with discriminant equal to .
Which odd primes are properly represented by some positive definite binary quadratic form (with integer coefficients) of discriminant ?
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3.I.1H
2008 commentProve that, for all , we have
[You may assume that, for ,
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3.II.11H
2008 commentState the reciprocity law for the Jacobi symbol.
Let be an odd integer , which is not a square. Prove that there exists a positive integer such that and
Prove further that there exist infinitely many prime numbers such that
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4.I.1H
2008 commentLet be an odd prime number. Assuming that the multiplicative group of is cyclic, prove that the multiplicative group of units of is cyclic for all .
Find an integer such that its residue class in generates the multiplicative group of units for all .
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4.II.11H
2008 commentLet be an integer, which is not a square, and let be the convergents to . Prove that
Explain briefly how this result can be used to generate a factor base , and a set of -numbers which may lead to a factorization of .
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1.I.2F
2008 commentLet be non-zero orthogonal polynomials on an interval such that the degree of is equal to for every , where the orthogonality is with respect to the inner product . If is any continuous function on orthogonal to and not identically zero, prove that must have at least distinct zeros in .
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2.II.11F
2008 commentLet be an operator satisfying the conditions
(i) for any with ,
(ii) for any and and
(iii) for any , where denotes the set of zeros of .
Prove that there exists a function with such that for every .
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2008
comment(a) State the Baire category theorem in its closed sets version.
(b) Let be a continuous function for each and suppose that there is a function such that for each . Prove that for each , there exists an integer and a non-empty open interval such that for all and .
[Hint: consider, for , the sets
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2008
comment(a) State Brouwer's fixed point theorem in the plane and prove that the statement is equivalent to non-existence of a continuous retraction of the closed disk to its boundary .
(b) Use Brouwer's fixed point theorem to prove that there is a complex number in the closed unit disc such that .
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2008
comment(a) State Liouville's theorem on approximation of algebraic numbers by rationals.
(b) Consider the continued fraction expression
in which the coefficients are all positive integers forming an unbounded set. Let be the th convergent. Prove that
and use this inequality together with Liouville's theorem to deduce that is irrational.
[ You may assume without proof that, for ,
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4.I.2F
2008 comment(a) State Runge's theorem on uniform approximation of analytic functions by polynomials.
(b) Suppose is analytic on
Prove that there exists a sequence of polynomials which converges to uniformly on compact subsets of .
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1.I.3G
2008 commentProve that an isometry of Euclidean space is an affine transformation.
Deduce that a finite group of isometries of has a common fixed point.
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1.II.11G
2008 commentWhat is meant by an inversion in a circle in ? Show that a composition of two inversions is a Möbius transformation.
Hence, or otherwise, show that if and are two disjoint circles in , then the composition of the inversions in and has two fixed points.
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2.I.3G
2008 commentState a theorem classifying lattices in . Define a frieze group.
Show there is a frieze group which is isomorphic to but is not generated by a translation, and draw a picture whose symmetries are this group.
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3.I.3G
2008 commentLet denote the Hausdorff dimension of a set in . Prove that if then is totally disconnected.
[You may assume that if is a Lipschitz map then
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4.I
2008 commentDefine the hyperbolic metric (in the sense of metric spaces) on the 3 -ball.
Given a finite set in hyperbolic 3 -space, show there is at least one closed ball of minimal radius containing that set.
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4.II.12G
2008 commentWhat does it mean for a subgroup of the Möbius group to be discrete?
Show that a discrete group necessarily acts properly discontinuously in hyperbolic 3-space.
[You may assume that a discrete subgroup of a matrix group is a closed subset.]
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1.I.4G
2008 commentDefine the entropy of a random variable that takes no more than different values. What are the maximum and the minimum values for the entropy for a fixed value of ? Explain when the maximum and minimum are attained. You should prove any inequalities that you use.
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2.I.4G
2008 commentDescribe briefly the Shannon-Fano and Huffman binary codes for a finite alphabet. Find examples of such codes for the alphabet when the four letters are taken with probabilities and respectively.
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1.II.12G
2008 commentState Shannon's Noisy Coding Theorem for a binary symmetric channel.
Define the mutual information of two discrete random variables and . Prove that the mutual information is symmetric and non-negative. Define also the information capacity of a channel.
A channel transmits numbers chosen from the alphabet and has transition matrix
for a number with . Calculate the information capacity of the channel.
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3.I.4G
2008 commentDefine the Hamming code and prove that the minimum distance between two distinct code words is 3. Explain how the Hamming code allows one error to be corrected.
A new code is obtained by using the Hamming code for the first 7 bits and taking the last bit as a check digit on the previous 7 . Find the minimum distance between two distinct code words for this code. How many errors can this code detect? How many errors can it correct?
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2.II.12G
2008 commentDescribe the Rabin cipher with modulus , explaining how it can be deciphered by the intended recipient and why it is difficult for an interceptor to decipher it.
The Bursars' Committee decides to communicate using Rabin ciphers to maintain confidentiality. The secretary of the committee encrypts a message, thought of as a positive integer , using the Rabin cipher with modulus (with ) and publishes both the encrypted message and the modulus. A foolish bursar deciphers this message to read it but then encrypts it again using a Rabin cipher with a different modulus (with and publishes the newly encrypted message and . The president of CUSU, who happens to be a talented mathematician, knows that this has happened. Explain how the president can work out what the original message was using the two different encrypted versions.
Can the president of CUSU also decipher other messages sent out by the Bursars' Committee?
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4.I.4G
2008 commentWhat is a binary cyclic code of length ? What is the generator polynomial for such a cyclic code? Prove that the generator polynomial is a factor of over the field .
Find all the binary cyclic codes of length 5 .
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1.I.5J
2008 commentConsider the following Binomial generalized linear model for data , with logit link function. The data are regarded as observed values of independent random variables , where
where is an unknown -dimensional parameter, and where are known dimensional explanatory variables. Write down the likelihood function for under this model.
Show that the maximum likelihood estimate satisfies an equation of the form , where is the matrix with rows , and where , with a function of and , which you should specify.
Define the deviance and find an explicit expression for in terms of and in the case of the model above.
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1.II.13J
2008 commentConsider performing a two-way analysis of variance (ANOVA) on the following data:

Explain and interpret the R commands and (slightly abbreviated) output below. In particular, you should describe the model being fitted, and comment on the hypothesis tests which are performed under the summary and anova commands.
[1]
as.vector
, length
, length
Coefficients:
Estimate Std. Error t value
(Intercept)
b2
b3

The following code fits a similar model. Briefly explain the difference between this model and the one above. Based on the output of the anova call below, say whether you prefer this model over the one above, and explain your preference.

Finally, explain what is being calculated in the code below and give the value that would be obtained by the final line of code.

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2008
commentSuppose that we want to estimate the angles and (in radians, say) of the triangle , based on a single independent measurement of the angle at each corner. Suppose that the error in measuring each angle is normally distributed with mean zero and variance . Thus, we model our measurements as the observed values of random variables
where are independent, each with distribution . Find the maximum likelihood estimate of based on these measurements.
Can the assumption that be criticized? Why or why not?
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3.I.5J
2008 commentConsider the linear model . Here, is an -dimensional vector of observations, is a known matrix, is an unknown -dimensional parameter, and , with unknown. Assume that has full rank and that . Suppose that we are interested in checking the assumption . Let , where is the maximum likelihood estimate of . Write in terms of an expression for the projection matrix which appears in the maximum likelihood equation .
Find the distribution of , and show that, in general, the components of are not independent.
A standard procedure used to check our assumption on is to check whether the studentized fitted residuals
look like a random sample from an distribution. Here,
Say, briefly, how you might do this in R.
This procedure appears to ignore the dependence between the components of noted above. What feature of the given set-up makes this reasonable?
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4.I
2008 commentA long-term agricultural experiment had grassland plots, each , differing in biomass, soil pH, and species richness (the count of species in the whole plot). While it was well-known that species richness declines with increasing biomass, it was not known how this relationship depends on soil pH. In the experiment, there were 30 plots of "low pH", 30 of "medium pH" and 30 of "high pH". Three lines of the data are reproduced here as an aid.
Briefly explain the commands below. That is, explain the models being fitted.
Let and denote the hypotheses represented by the three models and fits. Based on the output of the code below, what hypotheses are being tested, and which of the models seems to give the best fit to the data? Why?

Finally, what is the value obtained by the following command?

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4.II.13J
2008 commentConsider the following generalized linear model for responses as a function of explanatory variables , where for . The responses are modelled as observed values of independent random variables , with
Here, is a given link function, and are unknown parameters, and the are treated as known.
[Hint: recall that we write to mean that has density function of the form
for given functions a and
[ You may use without proof the facts that, for such a random variable ,
Show that the score vector and Fisher information matrix have entries:
and
How do these expressions simplify when the canonical link is used?
Explain briefly how these two expressions can be used to obtain the maximum likelihood estimate for .
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1.I.6B
2008 commentA gene product with concentration is produced by a chemical of concentration , is autocatalysed and degrades linearly according to the kinetic equation
where is a constant.
First consider the case . Show that if there are two positive steady states, and determine their stability. Sketch the reaction rate .
Now consider . Show that there is a single steady state if exceeds a critical value. If the system starts in the steady state with and then is increased sufficiently before decreasing back to zero, show that a biochemical switch can be achieved to a state , whose value you should determine.
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2.I.6B
2008 commentThe population dynamics of a species is governed by the discrete model
where and are positive constants.
Determine the steady states and their eigenvalues. Show that a period-doubling bifurcation occurs at .
Show graphically that the maximum possible population after is
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2.II.13B
2008 commentConsider the nonlinear equation describing the invasion of a population
with and a constant.
(a) Considering time-dependent spatially homogeneous solutions, show that there are two stable and one unstable uniform steady states.
(b) In the case , find the stationary 'front' which has
[Hint: where .]
(c) Now consider travelling-wave solutions to (1) of the form where . Show that satisfies an equation of the form
where and ().
Sketch the form of for and . Using conditions (2), show that
Deduce how the sign of the travelling-wave velocity depends on .
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3.I.6B
2008 commentAn allosteric enzyme reacts with a substrate to produce a product according to the mechanism
where and are enzyme-substrate complexes. With lowercase letters denoting concentrations, write down a system of differential equations based on the Law of Mass Action which model this reaction mechanism.
The initial conditions are Using , and , show that the nondimensional reaction mechanism reduces to
finding expressions for and .
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3.II.13B
2008 commentConsider the activator-inhibitor system in the fast-inhibitor limit
where is small, and .
Examine the linear stability of the state using perturbations of the form . Sketch the growth-rate as a function of the wavenumber . Find the growth-rate of the most unstable wave, and so determine the boundary in the - parameter plane which separates stable and unstable modes.
Show that the system is unchanged under the transformation and . Hence write down the equation for the boundary between stable and unstable modes of the state .
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4.I.6B
2008 commentA semi-infinite elastic filament lies along the positive -axis in a viscous fluid. When it is perturbed slightly to the shape , it evolves according to
where characterises the viscous drag and the bending stiffness. Motion is forced by boundary conditions
Use dimensional analysis to find the characteristic length of the disturbance. Show that the steady oscillating solution takes the form
finding the ordinary differential equation for .
Find two solutions for which decay as . Without solving explicitly for the amplitudes, show that is the superposition of two travelling waves which decay with increasing , one with crests moving to the left and one to the right. Which dominates?
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1.I.7A
2008 commentSketch the phase plane of the system
(i) for and (ii) for . Include in your sketches any trajectories that are the separatrices of a saddle point. In case (ii) shade the domain of stability of the origin.
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3.II.14A
2008 commentDefine the Poincaré index of a simple closed curve, not necessarily a trajectory, and the Poincaré index of an isolated fixed point for a dynamical system in . State the Poincaré index of a periodic orbit.
Consider the system
where and are constants and .
(a) Find and classify the fixed points, and state their Poincaré indices.
(b) By considering a suitable function , show that any periodic orbit satisfies
where is evaluated along the orbit.
(c) Deduce that if then the second-order differential equation
has no periodic solutions.
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2.I.7A
2008 commentExplain the difference between a stationary bifurcation and an oscillatory bifurcation for a fixed point of a dynamical system in with a real parameter .
The normal form of a Hopf bifurcation in polar coordinates is
where and are constants, , and . Sketch the phase plane near the bifurcation for each of the cases (i) , (ii) , (iii) and (iv) .
Let be the radius and the period of the limit cycle when one exists. Sketch how varies with for the case when the limit cycle is subcritical. Find the leading-order approximation to for .
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4.II.14A
2008 commentExplain the difference between a hyperbolic and a nonhyperbolic fixed point for a dynamical system in .
Consider the system in , where is a real parameter,
Show that the fixed point has a bifurcation when , while the fixed points have a bifurcation when .
[The fixed point at should not be considered further.]
Analyse each of the bifurcations at and in turn as follows. Make a change of variable of the form . Identify the (non-extended) stable and centre linear subspaces at the bifurcation in terms of and . By finding the leading-order approximation to the extended centre manifold, construct the evolution equation on the extended centre manifold, and determine the type of bifurcation. Sketch the local bifurcation diagram, showing which fixed points are stable.
[Hint: the leading-order approximation to the extended centre manifold of the bifurcation at is for some coefficient a.]
Show that there is another fixed point in for , and that this fixed point connects the two bifurcations.
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3.I.7A
2008 commentState the normal-form equations for (i) a saddle-node bifurcation, (ii) a transcritical bifurcation and (iii) a pitchfork bifurcation, for a one-dimensional map .
Consider a period-doubling bifurcation of the form
where , and as . Show that
where , and the parameters and are to be identified in terms of , and . Deduce the condition for the bifurcation to be supercritical.
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4.I.7A
2008 commentLet be a continuous one-dimensional map of an interval . State when is chaotic according to Glendinning's definition.
Prove that if has a 3 -cycle then has a horseshoe.
[You may assume the Intermediate Value Theorem.]
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1.I.8C
2008 commentThe function is defined by
For which values of does the integral converge?
Show that, for these values,
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2.I.8C
2008 commentThe Beta function is defined for by
and by analytic continuation elsewhere in the complex -plane.
Show that
and explain how this result can be used to obtain the analytic continuation of . Hence show that is analytic except for simple poles and find the residues at the poles.
-
3.I.8C
2008 commentWhat is the effect of the Möbius transformation on the points , and ?
By considering
or otherwise, show that is a branch of the -function

Give a linearly independent branch.
-
1.II.14C
2008 commentShow that under the change of variable the equation
becomes
Show that this is a Papperitz equation and that the corresponding -function is
Deduce that .
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4.I.8C
2008 commentThe Hilbert transform of a function is defined by
where denotes the Cauchy principal value.
Show that the Hilbert transform of is
-
2.II.14C
2008 comment(i) The function is defined by
where is the circle , described anti-clockwise starting on the positive real axis and where the value of at each point on is determined by analytic continuation along with at the starting point. Verify by direct integration that is an entire function, the values of which depend on .
(ii) The function is defined by
where is a figure of eight, starting at , looping anti-clockwise round and returning to , then looping clockwise round and returning again to . The value of is determined by analytic continuation along with at the start. Show that, for ,
where
Explain how this provides the analytic continuation of . Classify the singular points of the analytically continued function, commenting on the points .
Explain briefly why the analytic continuation could not be obtained by this method if were replaced by the circle .
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1.I.9A
2008 commentThe action for a system with generalized coordinates for a time interval is given by
where is the Lagrangian. The end point values and are fixed.
Derive Lagrange's equations from the principle of least action by considering the variation of for all possible paths.
Define the momentum conjugate to . Derive a condition for to be a constant of the motion.
A symmetric top moves under the action of a potential . The Lagrangian is given by
where the generalized coordinates are the Euler angles and the principal moments of inertia are and .
Show that is a constant of the motion and give expressions for two others. Show further that it is possible for the top to move with both and constant provided these satisfy the condition
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2.II.15B
2008 commentA particle of mass , charge and position vector moves in a magnetic field whose vector potential is A. Its Hamiltonian is given by
Write down Hamilton's equations and use them to derive the equations of motion for the charged particle.
Define the Poisson bracket for general and . Show that for motion governed by the above Hamiltonian
Consider the vector potential to be given by , where . Use Hamilton's equations to show that is constant and that circular motion at radius with angular frequency is possible provided that
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2.I.9A
2008 commentA system of particles , with mass , moves around a circle of radius . The angle between the radius to particle and a fixed reference radius is . The interaction potential for the system is
where is a constant and .
The Lagrangian for the system is
Write down the equation of motion for particle and show that the system is in equilibrium when the particles are equally spaced around the circle.
Show further that the system always has a normal mode of oscillation with zero frequency. What is the form of the motion associated with this?
Find all the frequencies and modes of oscillation when and , where is a constant.
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3.I.9E
2008 commentWriting , Hamilton's equations may be written in the form
where the matrix
and and 0 denote the unit and zero matrices respectively.
Explain what is meant by the statement that the transformation ,
is canonical, and show that the condition for this is that
where is the Jacobian matrix with elements
Use this condition to show that for a system with the transformation given by
is canonical.
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4.II.15B
2008 comment(a) A Hamiltonian system with degrees of freedom has Hamiltonian , where the coordinates and the momenta respectively.
Show from Hamilton's equations that when does not depend on time explicitly, for any function ,
where denotes the Poisson bracket.
For a system of interacting vortices
where is a constant. Show that the quantity defined by
is a constant of the motion.
(b) The action for a Hamiltonian system with one degree of freedom with for which the motion is periodic is
Show without assuming any specific form for that the period of the motion is given by
Suppose now that the system has a parameter that is allowed to vary slowly with time. Explain briefly what is meant by the statement that the action is an adiabatic invariant. Suppose that when this parameter is fixed, when . Deduce that, if decreases on an orbit with any when the parameter is slowly varied, then increases.
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4.I.9B
2008 comment(a) Show that the principal moments of inertia for an infinitesimally thin uniform rectangular sheet of mass with sides of length and (with ) about its centre of mass are and .
(b) Euler's equations governing the angular velocity of the sheet as viewed in the body frame are
and
A possible solution of these equations is such that the sheet rotates with , and constant.
By linearizing, find the equations governing small motions in the neighbourhood of this solution that have . Use these to show that there are solutions corresponding to instability such that and are both proportional to exp , with
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1.I.10E
2008 commentThe number density of particles of mass at equilibrium in the early universe is given by the integral
where is the chemical potential, and is the spin degeneracy. Assuming that the particles remain in equilibrium when they become non-relativistic , show that the number density can be expressed as
[Hint: Recall that
At around seconds, deuterium forms through the nuclear fusion of nonrelativistic protons and neutrons via the interaction . In equilibrium, what is the relationship between the chemical potentials of the three species? Show that the ratio of their number densities can be expressed as
where the deuterium binding energy is and you may take . Now consider the fractional densities , where is the baryon density of the universe, to re-express the ratio above in the form , which incorporates the baryon-to-photon ratio of the universe.
[You may assume that the photon density is .]
Why does deuterium form only at temperatures much lower than that given by ?
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2.I.10E
2008 commentA spherically-symmetric star obeys the pressure-support equation
where is the pressure at a distance from the centre, is the density, and is the mass within a sphere of radius . Show that this implies
Propose and justify appropriate boundary conditions for the pressure at the centre of the star and at its outer edge .
Show that the function
is a decreasing function of . Deduce that the central pressure satisfies
where is the mass of the star.
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1.II.15E
2008 comment(i) A homogeneous and isotropic universe has mass density and scale factor . Show how the conservation of total energy (kinetic plus gravitational potential) when applied to a test particle on the edge of a spherical region in this universe can be used to obtain the Friedmann equation
where is a constant. State clearly any assumptions you have made.
(ii) Assume that the universe is flat and filled with two major components: pressure-free matter and dark energy with equation of state where their mass densities today are given respectively by and . Assuming that each component independently satisfies the fluid conservation equation, , show that the total mass density can be expressed as
where we have set .
Hence, solve the Friedmann equation and show that the scale factor can be expressed in the form
where and are constants which you should specify in terms of and .
[Hint: try the substitution .]
Show that the scale factor has the expected behaviour for a matter-dominated universe at early times and that the universe accelerates at late times .
-
2008
commentThe energy density and pressure of photons in the early universe is given by
where is the Stefan-Boltzmann constant. By using the first law of thermodynamics , deduce that the entropy differential can be expressed in the form
With the third law, show that the entropy density is given by .
While particle interaction rates remain much greater than the Hubble parameter , justify why entropy will be conserved during the expansion of the universe. Hence, in the early universe (radiation domination) show that the temperature where is the scale factor of the universe, and show that the Hubble parameter .
-
4.I.10E
2008 commentThe Friedmann and Raychaudhuri equations are respectively
where is the mass density, is the pressure, is the curvature and with the cosmic time. Using conformal time (defined by ) and the equation of state , show that these can be rewritten as
where and the relative density is .
Use these relations to derive the following evolution equation for
For both and , plot the qualitative evolution of as a function of in an expanding universe (i.e. include curves initially with and ).
Hence, or otherwise, briefly describe the flatness problem of the standard cosmology and how it can be solved by inflation.
-
2008
commentSmall density perturbations in pressureless matter inside the cosmological horizon obey the following Fourier evolution equation
where is the average background density of the pressureless gravitating matter and is the comoving wavevector.
(i) Seek power law solutions constant) during the matter-dominated epoch to find the approximate solution
where are functions of only and is the time of equal matter-radiation.
By considering the behaviour of the scalefactor and the relative density , show that early in the radiation era there is effectively no significant perturbation growth in on sub-horizon scales.
(ii) For a given wavenumber , show that the time at which this mode crosses inside the horizon, i.e., , is given by
where , and the equal matter-radiation redshift is given by .
Assume that primordial perturbations from inflation are scale-invariant with a constant amplitude as they cross the Hubble radius given by , where is a constant and is a large volume. Use the results of (i) to project these perturbations forward to , and show that the power spectrum for perturbations today will be given approximately by
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1.II.16G
2008 commentWhat is a well-ordered set? Show that given any two well-ordered sets there is a unique order isomorphism between one and an initial segment of the other.
Show that for any ordinal and for any non-zero ordinal there are unique ordinals and with and .
Show that a non-zero ordinal is a limit ordinal if and only if for some non-zero ordinal .
[You may assume standard properties of ordinal addition, multiplication and subtraction.]
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2.II.16G
2008 comment(i) State the Completeness Theorem and the Compactness Theorem for the predicate calculus.
(ii) Show that if a theory has arbitrarily large finite models then it has an infinite model. Deduce that there is no first order theory whose models are just the finite fields of characteristic 2 . Show that the theory of infinite fields of characteristic 2 does not have a finite axiomatisation.
(iii) Let be the collection of closed terms in some first order language . Suppose that is a provable sentence of with quantifier-free. Show that the set of sentences is inconsistent.
[Hint: consider the minimal substructure of a model.]
Deduce that there are in such that is provable.
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3.II.16G
2008 commentWhat is a transitive set? Show that if is transitive then so are the union and the power set of . If is transitive, is transitive? If is transitive, is transitive? Justify your answers.
What is the transitive closure of a set? Show that any set has a transitive closure .
Suppose that has rank . What is the rank of ? What is the rank of ?
[You may use standard properties of rank.]
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4.II.16G
2008 commentProve Hartog's Lemma that for any set there is an ordinal which cannot be mapped injectively into .
Now assume the Axiom of Choice. Prove Zorn's Lemma and the Well-ordering Principle.
[If you appeal to a fixed point theorem then you should state it clearly.]
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1.II .17F
2008 commentState a result of Euler concerning the number of vertices, edges and faces of a connected plane graph. Deduce that if is a planar graph then . Show that if is a planar graph then .
Are the following statements true or false? Justify your answers.
[You may quote standard facts about planar and non-planar graphs, provided that they are clearly stated.]
(i) If is a graph with then is planar.
(ii) If is a connected graph with average degree at most then is planar.
(iii) If is a connected graph with average degree at most 2 then is planar.
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2.II.17F
2008 commentProve that every graph on vertices with minimum degree is Hamiltonian. For each , give an example to show that this result does not remain true if we weaken the condition to (for even) or (for odd).
For any graph , let denote the graph formed by adding new vertices to , all joined to each other and to all vertices of . By considering , show that if is a graph on vertices with then has a Hamilton path (a path passing through all the vertices of ).
For each positive integer , exhibit a connected graph such that is not Hamiltonian. Is this still possible if we replace 'connected' with '2-connected'?
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3.II
2008 commentDefine the chromatic polynomial of a graph . Show that if has vertices and edges then
where and and for all . [You may assume the deletion-contraction relation, provided it is clearly stated.]
Show that if is a tree on vertices then . Does the converse hold?
[Hint: if is disconnected, how is the chromatic polynomial of related to the chromatic polynomials of its components?]
Show that if is a graph on vertices with the same chromatic polynomial as (the Turán graph on vertices with vertex classes) then must be isomorphic to .
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4.II.17F
2008 commentFor , let be the least integer such that for every 2 -colouring of the edges of there is a monochromatic . Prove that exists.
For any and , define the Ramsey number , and prove that it exists.
Show that, whenever the positive integers are partitioned into finitely many classes, some class contains with .
[Hint: given a finite colouring of the positive integers, induce a colouring of the pairs of positive integers by giving the pair the colour of .]
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1.II.18H
2008 commentFind the Galois group of the polynomial over (i) the finite field , (ii) the finite field , (iii) the finite field , (iv) the field of rational numbers.
[Results from the course which you use should be stated precisely.]
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2.II.18H
2008 comment(i) Let be a field, , and not divisible by the characteristic. Suppose that contains a primitive th root of unity. Show that the splitting field of has cyclic Galois group.
(ii) Let be a Galois extension of fields and denote a primitive th root of unity in some extension of , where is not divisible by the characteristic. Show that is a subgroup of .
(iii) Determine the minimal polynomial of a primitive 6 th root of unity over .
Compute the Galois group of .
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3.II.18H
2008 commentLet be a field extension.
(a) State what it means for to be algebraic over , and define its degree . Show that if is odd, then .
[You may assume any standard results.]
Show directly from the definitions that if are algebraic over , then so too is .
(b) State what it means for to be separable over , and for the extension to be separable.
Give an example of an inseparable extension .
Show that an extension is separable if is a finite field.
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4.II.18H
2008 commentLet be the function field in one variable, an integer, and .
Define by the formulae
and let be the group generated by and .
(i) Find such that .
[You must justify your answer, stating clearly any theorems you use.]
(ii) Suppose is an odd prime. Determine the subgroups of and the corresponding intermediate subfields , with .
State which intermediate subfields are Galois extensions of , and for these extensions determine the Galois group.
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1.II.19G
2008 commentFor a complex representation of a finite group , define the action of on the dual representation . If denotes the character of , compute the character of .
[Your formula should express just in terms of the character .]
Using your formula, how can you tell from the character whether a given representation is self-dual, that is, isomorphic to the dual representation?
Let be an irreducible representation of . Show that the trivial representation occurs as a summand of with multiplicity either 0 or 1 . Show that it occurs once if and only if is self-dual.
For a self-dual irreducible representation , show that either has a nondegenerate -invariant symmetric bilinear form or a nondegenerate -invariant alternating bilinear form, but not both.
If is an irreducible self-dual representation of odd dimension , show that the corresponding homomorphism is conjugate to a homomorphism into the orthogonal group . Here means the subgroup of that preserves a nondegenerate symmetric bilinear form on .
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2.II.19G
2008 commentA finite group of order 360 has conjugacy classes of sizes . The values of four of its irreducible characters are given in the following table.
Complete the character table.
[Hint: it will not suffice just to use orthogonality of characters.]
Deduce that the group is simple.
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3.II.19G
2008 commentLet denote the irreducible representation of ; thus has dimension 3. Compute the character of the representation of for any . Compute the dimension of the invariants , meaning the subspace of where acts trivially.
Hence, or otherwise, show that the ring of complex polynomials in three variables which are invariant under the action of is a polynomial ring. Find a generator for this polynomial ring.
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4.II.19G
2008 comment(a) Let be a normal subgroup of a finite group , and let be an irreducible representation of . Show that either restricted to is isotypic (a sum of copies of one irreducible representation of , or else is induced from an irreducible representation of some proper subgroup of .
(b) Using (a), show that every (complex) irreducible representation of a -group is induced from a 1-dimensional representation of some subgroup.
[You may assume that a nonabelian -group has an abelian normal subgroup which is not contained in the centre of .]
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1.II.20G
2008 comment(a) Define the ideal class group of an algebraic number field . State a result involving the discriminant of that implies that the ideal class group is finite.
(b) Put , where , and let be the ring of integers of . Show that . Factorise the ideals [2] and [3] in into prime ideals. Verify that the equation of ideals
holds. Hence prove that has class number 3 .
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2.II.20G
2008 comment(a) Factorise the ideals [2], [3] and [5] in the ring of integers of the field . Using Minkowski's bound
determine the ideal class group of .
[Hint: it might be helpful to notice that for some
(b) Find the fundamental unit of and determine all solutions of the equations in integers . Prove that there are in fact no solutions of in integers .
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4.II.20G
2008 comment(a) Explain what is meant by an integral basis of an algebraic number field. Specify such a basis for the quadratic field .
(b) Let with , a fourth root of 2 . Write an element of as
with . By computing the relative traces and , show that if is an algebraic integer of , then and are rational integers. By further computing the relative norm , show that
are rational integers. Deduce that is an integral basis of .
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1.II.21F
2008 comment(i) State the van Kampen theorem.
(ii) Calculate the fundamental group of the wedge .
(iii) Let where is a circle. Calculate the fundamental group of .
-
2.II.21F
2008 commentProve the Borsuk-Ulam theorem in dimension 2: there is no map such that for every . Deduce that is not homeomorphic to any subset of .
-
3.II.20F
2008 commentLet be the quotient space obtained by identifying one pair of antipodal points on . Using the Mayer-Vietoris exact sequence, calculate the homology groups and the Betti numbers of .
-
4.II.21F
2008 commentLet and be topological spaces.
(i) Show that a map is a homotopy equivalence if there exist maps such that and . More generally, show that a map is a homotopy equivalence if there exist maps such that and are homotopy equivalences.
(ii) Suppose that and are universal covering spaces of the path-connected, locally path-connected spaces and . Using path-lifting properties, show that if then .
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1.II.22F
2008 commentSuppose and are real numbers with and . Show, quoting any results on convexity that you need, that
for all real positive and .
Define the space and show that it is a complete normed vector space.
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2.II.22F
2008 commentState and prove the principle of uniform boundedness.
[You may assume the Baire category theorem.]
Suppose that and are Banach spaces. Suppose that
is linear and continuous in each variable separately, that is to say that, if is fixed,
is a continuous linear map and, if is fixed,
is a continuous linear map. Show that there exists an such that
for all . Deduce that is continuous.
Suppose and are Banach spaces. Suppose that
is linear and continuous in each variable separately. Does it follow that is continuous? Give reasons.
Suppose that and are Banach spaces. Suppose that
is continuous in each variable separately. Does it follow that is continuous? Give reasons.
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3.II.21F
2008 commentState and prove the Stone-Weierstrass theorem for real-valued functions. You may assume that the function can be uniformly approximated by polynomials on any interval .
Suppose that . Let be the set of functions which can be uniformly approximated on by polynomials with integer coefficients. By making appropriate use of the identity
or otherwise, show that .
Is it true that every continuous function on can be uniformly approximated by polynomials with integer coefficients?
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4.II.22F
2008 commentLet be a Hilbert space. Show that if is a closed subspace of then any can be written as with and .
Suppose is unitary (that is to say ). Let
and consider
(i) Show that is an isometry and .
(ii) Show that is a subspace of and as whenever .
(iii) Let be the closure of . Show that as whenever .
(iv) Show that, if , then . Deduce that, if , then .
(v) If show that there is a such that as .
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1.II.23H
2008 commentDefine the terms Riemann surface, holomorphic map between Riemann surfaces and biholomorphic map.
Show, without using the notion of degree, that a non-constant holomorphic map between compact connected Riemann surfaces must be surjective.
Let be a biholomorphic map of the punctured unit disc onto itself. Show that extends to a biholomorphic map of the open unit disc to itself such that .
Suppose that is a continuous holomorphic map between Riemann surfaces and is holomorphic on , where is a point in . Show that is then holomorphic on all of .
[The Open Mapping Theorem may be used without proof if clearly stated.]
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2.II.23H
2008 commentExplain what is meant by a divisor on a compact connected Riemann surface . Explain briefly what is meant by a canonical divisor. Define the degree of and the notion of linear equivalence between divisors. If two divisors on have the same degree must they be linearly equivalent? Give a proof or a counterexample as appropriate, stating accurately any auxiliary results that you require.
Define for a divisor , and state the Riemann-Roch theorem. Deduce that the dimension of the space of holomorphic differentials is determined by the genus of and that the same is true for the degree of a canonical divisor. Show further that if then admits a non-constant meromorphic function with at most two poles (counting with multiplicities).
[General properties of meromorphic functions and meromorphic differentials on may be used without proof if clearly stated.]
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3.II
2008 commentDefine the degree of a non-constant holomorphic map between compact connected Riemann surfaces and state the Riemann-Hurwitz formula.
Show that there exists a compact connected Riemann surface of any genus .
[You may use without proof any foundational results about holomorphic maps and complex algebraic curves from the course, provided that these are accurately stated. You may also assume that if is a non-constant complex polynomial without repeated roots then the algebraic curve is path connected.]
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4.II.23H
2008 commentLet be a lattice in generated by 1 and , where . The Weierstrass function is the unique meromorphic -periodic function on , such that the only poles of are at points of and as .
Show that is an even function. Find all the zeroes of .
Suppose that is a complex number such that . Show that the function
has no poles in . By considering the Laurent expansion of at , or otherwise, deduce that is constant.
[General properties of meromorphic doubly-periodic functions may be used without proof if accurately stated.]
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1.II.24H
2008 commentLet be an integer, and let denote the set of real-valued matrices. We make into an -dimensional smooth manifold via the obvious identification .
(a) Let denote the subset
Show that is a submanifold of . What is ?
(b) Now let denote the subset
Show that for ,
Show that is a submanifold of . What is the dimension of
(c) Now consider the set . For what values of is a submanifold of ?
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2.II
2008 comment(a) For a regular curve in , define curvature and torsion and state the Frenet formulas.
(b) State and prove the isoperimetric inequality for domains with compact closure and boundary .
[You may assume Wirtinger's inequality.]
(c) Let be a closed plane regular curve such that is contained in a disc of radius . Show that there exists such that , where denotes the signed curvature. Show by explicit example that the assumption of closedness is necessary.
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3.II.23H
2008 commentLet be a surface.
(a) Define the Gauss Map, principal curvatures , Gaussian curvature and mean curvature . State the Theorema Egregium.
(b) Define what is meant for to be minimal. Prove that if is minimal, then . Give an example of a minimal surface whose Gaussian curvature is not identically 0 , justifying your answer.
(c) Does there exist a compact minimal surface ? Justify your answer.
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4.II
2008 commentLet be a surface.
(a) In the case where is compact, define the Euler characteristic and genus of .
(b) Define the notion of geodesic curvature for regular curves . When is ? State the Global Gauss-Bonnet Theorem (including boundary term).
(c) Let (the standard 2-sphere), and suppose is a simple closed regular curve such that is the union of two distinct connected components with equal areas. Can have everywhere strictly positive or everywhere strictly negative geodesic curvature?
(d) Prove or disprove the following statement: if is connected with Gaussian curvature identically, then is a subset of a sphere of radius 1 .
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1.II
2008 commentState the Dominated Convergence Theorem.
Hence or otherwise prove Kronecker's Lemma: if is a sequence of non-negative reals such that
then
Let be independent random variables and set . Let be the collection of all finite unions of intervals of the form , where and are rational, together with the whole line . Prove that with probability 1 the limit
exists for all , and identify it. Is it possible to extend defined on to a measure on the Borel -algebra of ? Justify your answer.
-
2.II
2008 commentExplain what is meant by a simple function on a measurable space .
Let be a finite measure space and let be a non-negative Borel measurable function. State the definition of the integral of with respect to .
Prove that, for any sequence of simple functions such that for all , we have
State and prove the Monotone Convergence Theorem for finite measure spaces.
-
2008
comment(i) What does it mean to say that a sequence of random variables converges in probability to ? What does it mean to say that the sequence converges in distribution to ? Prove that if in probability, then in distribution.
(ii) What does it mean to say that a sequence of random variables is uniformly integrable? Show that, if is uniformly integrable and in distribution, then .
[Standard results from the course may be used without proof if clearly stated.]
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2.II.26I
2008 commentConsider a continuous-time Markov chain given by the diagram below.

We will assume that the rates and are all positive.
(a) Is the chain irreducible?
(b) Write down the standard equations for the hitting probabilities
and
Explain how to identify the probabilities and among the solutions to these equations.
[You should state the theorem you use but its proof is not required.]
(c) Set and find a matrix such that
The recursion matrix has a 'standard' eigenvalue and a 'standard' eigenvector that do not depend on the transition rates: what are they and why are they always present?
(d) Calculate the second eigenvalue of the matrix , and the corresponding eigenvector, in the form , where .
(e) Suppose the second eigenvalue is . What can you say about and ? Is the chain transient or recurrent? Justify your answer.
(f) Now assume the opposite: the second eigenvalue is . Check that in this case . Is the chain transient or recurrent under this condition?
(g) Finally, specify, by means of inequalities between the parameters and , when the chain is recurrent and when it is transient.
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4.II
2008 comment(i) A stepfunction is any function on which can be written in the form
where are real numbers, with for all . Show that the set of all stepfunctions is dense in . Here, denotes the Borel -algebra, and denotes Lebesgue measure.
[You may use without proof the fact that, for any Borel set of finite measure, and any , there exists a finite union of intervals such that .]
(ii) Show that the Fourier transform
of a stepfunction has the property that as .
(iii) Deduce that the Fourier transform of any integrable function has the same property.
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1.II.26I
2008 commentLet be an irreducible continuous-time Markov chain with initial probability distribution and Q-matrix (for short: a CTMC), on a finite state space .
(i) Define the terms reversible CTMC and detailed balance equations (DBEs) and explain, without proof, the relation between them.
(ii) Prove that any solution of the DBEs is an equilibrium distribution (ED) for .
Let be an irreducible discrete-time Markov chain with initial probability distribution and transition probability matrix (for short: a DTMC), on the state space .
(iii) Repeat the two definitions from (i) in the context of the DTMC . State also in this context the relation between them, and prove a statement analogous to (ii).
(iv) What does it mean to say that is the jump chain for ? State and prove a relation between the ED for the and the ED for its jump chain .
(v) Prove that is reversible (in equilibrium) if and only if its jump chain is reversible (in equilibrium).
(vi) Consider now a continuous time random walk on a graph. More precisely, consider a CTMC on an undirected graph, where some pairs of states are joined by one or more non-oriented 'links' . Here is the number of links between and . Assume that the jump rate is proportional to . Can the chain be reversible? Identify the corresponding jump chain (which determines a discrete-time random walk on the graph) and comment on its reversibility.
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3.II.25I
2008 commentLet be an irreducible continuous-time Markov chain with countably many states. What does it mean to say the chain is (i) positive recurrent, (ii) null recurrent? Consider the chain with the arrow diagram below.

In this question we analyse the existence of equilibrium probabilities and of the chain being in state or , and the impact of this fact on positive and null recurrence of the chain.
(a) Write down the invariance equations and check that they have the form
where is a recursion matrix:
(b) Verify that the row vector is an eigenvector of with the eigenvalue where
Hence, specify the form of equilibrium probabilities and and conclude that the chain is positive recurrent if and only if .
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4.II.26I
2008 commentOn a hot summer night, opening my window brings some relief. This attracts hordes of mosquitoes who manage to negotiate a dense window net. But, luckily, I have a mosquito trapping device in my room.
Assume the mosquitoes arrive in a Poisson process at rate ; afterwards they wander around for independent and identically distributed random times with a finite mean , where denotes the random wandering time of a mosquito, and finally are trapped by the device.
(a) Identify a mathematical model, which was introduced in the course, for the number of mosquitoes present in the room at times .
(b) Calculate the distribution of in terms of and the tail probabilities of the wandering time , where is the number of mosquitoes in the room at time (assuming that at the initial time, ).
(c) Write down the distribution for , the number of mosquitoes in the room in equilibrium, in terms of and .
(d) Instead of waiting for the number of mosquitoes to reach equilibrium, I close the window at time . For let be the number of mosquitoes left at time , i.e. time units after closing the window. Calculate the distribution of .
(e) Let be the time needed to trap all mosquitoes in the room after closing the window at time . By considering the event , or otherwise, compute .
(f) Now suppose that the time at which I shut the window is very large, so that I can assume that the number of mosquitoes in the room has the distribution of . Let be the further time needed to trap all mosquitoes in the room. Show that
where .
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1.II.27I
2008 commentAn angler starts fishing at time 0. Fish bite in a Poisson Process of rate per hour, so that, if , the number of fish he catches in the first hours has the Poisson distribution , while , the time in hours until his th bite, has the Gamma distribution , with density function
Bystander plans to watch for 3 hours, and to record the number of fish caught. Bystander plans to observe until the 10 th bite, and to record , the number of hours until this occurs.
For , show that is an unbiased estimator of whose variance function achieves the Cramér-Rao lower bound
Find an unbiased estimator of for , of the form . Does it achieve the Cramér-Rao lower bound? Is it minimum-variance-unbiased? Justify your answers.
In fact, the 10 th fish bites after exactly 3 hours. For each of and , write down the likelihood function for based their observations. What does the Likelihood Principle have to say about the inferences to be drawn by and , and why? Compute the estimates and produced by applying and to the observed data. Does the method of minimum-variance-unbiased estimation respect the Likelihood Principle?
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2.II.27I
2008 commentUnder hypothesis , a real-valued observable , taking values in , has density function . Define the Type I error and the Type II error of a test of the null hypothesis against the alternative hypothesis . What are the size and power of the test in terms of and ?
Show that, for minimises among all possible tests if and only if it satisfies
What does this imply about the admissibility of such a test?
Given the value of a parameter variable , the observable has density function
For fixed , describe all the likelihood ratio tests of against .
For fixed , let be the test that rejects if and only if . Is admissible as a test of against for every ? Is it uniformly most powerful for its size for testing against the composite hypothesis ? Is it admissible as a test of against ?
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3.II.26I
2008 commentDefine the notion of exponential family , and show that, for data arising as a random sample of size from an exponential family, there exists a sufficient statistic whose dimension stays bounded as .
The log-density of a normal distribution can be expressed in the form
where is the value of an unknown parameter . Determine the function , and the natural parameter-space . What is the mean-value parameter in terms of
Determine the maximum likelihood estimator of based on a random sample , and give its asymptotic distribution for .
How would these answers be affected if the variance of were known to have value ?
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4.II.27I
2008 commentDefine sufficient statistic, and state the factorisation criterion for determining whether a statistic is sufficient. Show that a Bayesian posterior distribution depends on the data only through the value of a sufficient statistic.
Given the value of an unknown parameter , observables are independent and identically distributed with distribution . Show that the statistic is sufficient for .
If the prior distribution is , determine the posterior distribution of and the predictive distribution of .
In fact, there are two hypotheses as to the value of M. Under hypothesis , takes the known value 0 ; under is unknown, with prior distribution . Explain why the Bayes factor for choosing between and depends only on , and determine its value for data .
The frequentist -level test of against rejects when . What is the Bayes factor for the critical case ? How does this behave as ? Comment on the similarities or differences in behaviour between the frequentist and Bayesian tests.
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1.II.28J
2008 comment(a) In the context of the Black-Scholes formula, let be the time- 0 spot price, be the strike price, be the time to maturity, and let be the volatility. Assume that the interest rate is constant and assume absence of dividends. Write for the time- 0 price of a standard European call. The Black-Scholes formula can be written in the following form
State how the quantities and depend on and .
Assume that you sell this option at time 0 . What is your replicating portfolio at time 0 ?
[No proofs are required.]
(b) Compute the limit of as . Construct an explicit arbitrage under the assumption that European calls are traded above this limiting price.
(c) Compute the limit of as . Construct an explicit arbitrage under the assumption that European calls are traded below this limiting price.
(d) Express in terms of and the derivative
[Hint: you may find it useful to express in terms of .]
[You may use without proof the formula .]
(e) Say what is meant by implied volatility and explain why the previous results make it well-defined.
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2.II.28J
2008 comment(a) Let be a Brownian motion and consider the process
for deterministic. For which values of is a supermartingale? For which values of is a martingale? For which values of is a martingale? Justify your answers.
(b) Assume that the riskless rates of return for Dollar investors and Euro investors are and respectively. Thus, 1 Dollar at time 0 in the bank account of a Dollar investor will grow to Dollars at time . For a Euro investor, the Dollar is a risky, tradable asset. Let be his equivalent martingale measure and assume that the EUR/USD exchange rate at time , that is, the number of Euros that one Dollar will buy at time , is given by
where is a Brownian motion under . Determine as function of and . Verify that is a martingale if .
(c) Let be as in part (b). Let now be an equivalent martingale measure for a Dollar investor and assume that the EUR/USD exchange rate at time is given by
where now is a Brownian motion under . Determine as function of and . Given , check, under , that is is not a martingale but that is a martingale.
(d) Assuming still that , rederive the final conclusion of part (c), namely the martingale property of , directly from part (b).
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3.II.27J
2008 commentConsider a vector of asset prices evolving over time . The asset price is assumed constant over time. In this context, explain what is an arbitrage and prove that the existence of an equivalent martingale measure implies noarbitrage.
Suppose that over two periods a stock price moves on a binomial tree

Assume riskless rate . Determine the equivalent martingale measure. [No proof is required.]
Sell an American put with strike 15 and expiry 2 at its no-arbitrage price, which you should determine.
Verify that the buyer of the option should use his early exercise right if the first period is bad.
Assume that the first period is bad, and that the buyer forgets to exercise. How much risk-free profit can you lock in?
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4.II.28J
2008 comment(a) Consider a family of independent, identically distributed, positive random variables and fix . Define inductively
Compute, for , the conditional expectation .
(b) Fix . In the setting of part (a), compute also , where
(c) Let be as in part (b). An investor with wealth at time 0 wishes to invest it in such a way as to maximise where is the wealth at the start of day . Let be fixed. On day , he chooses the proportion of his wealth to invest in a single risky asset, so that his wealth at the start of day will be
Here, is as in part (a) and is the per-period riskless rate of interest. If denotes the value function of this optimization problem, show that and give a formula for . Verify that, in the case , your answer is in agreement with part (b).
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2.II.29I
2008 commentConsider a stochastic controllable dynamical system with action-space and countable state-space . Thus and denotes the transition probability from to when taking action . Suppose that a cost is incurred each time that action is taken in state , and that this cost is uniformly bounded. Write down the dynamic optimality equation for the problem of minimizing the expected long-run average cost.
State in terms of this equation a general result, which can be used to identify an optimal control and the minimal long-run average cost.
A particle moves randomly on the integers, taking steps of size 1 . Suppose we can choose at each step a control parameter , where is fixed, which has the effect that the particle moves in the positive direction with probability and in the negative direction with probability . It is desired to maximize the long-run proportion of time spent by the particle at 0 . Show that there is a solution to the optimality equation for this example in which the relative cost function takes the form , for some constant .
Determine an optimal control and show that the maximal long-run proportion of time spent at 0 is given by
You may assume that it is valid to use an unbounded function in the optimality equation in this example.
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3.II.28I
2008 commentLet be a positive-definite symmetric matrix. Show that a non-negative quadratic form on of the form
is minimized over , for each , with value , by taking , where .
Consider for the controllable stochastic linear system in
starting from at time , where the control variables take values in , and where are independent, zero-mean random variables, with . Here, and are, respectively, and matrices. Assume that a cost is incurred at each time and that a final cost is incurred at time . Here, is a given non-negative-definite symmetric matrix. It is desired to minimize, over the set of all controls , the total expected cost . Write down the optimality equation for the infimal cost function .
Hence, show that has the form
for some non-negative-definite symmetric matrix and some real constant . Show how to compute the matrix and constant and how to determine an optimal control.
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4.II.29I
2008 commentState Pontryagin's maximum principle for the controllable dynamical system with state-space , given by
where the running costs are given by , up to an unconstrained terminal time when the state first reaches 0 , and there is a terminal cost .
A company pays a variable price per unit time for electrical power, agreed in advance, which depends on the time of day. The company takes on a job at time , which requires a total amount of electrical energy, but can be processed at a variable level of power consumption . If the job is completed by time , then the company will receive a reward . Thus, it is desired to minimize
subject to
with unconstrained. Take as state variable the energy still needed at time to complete the job. Use Pontryagin's maximum principle to show that the optimal control is to process the job on full power or not at all, according as the price lies below or above a certain threshold value .
Show further that, if is the completion time for the optimal control, then
Consider a case in which is periodic, with period one day, where day 1 corresponds to the time interval , and during day 1 . Suppose also that and . Determine the total energy cost and the reward associated with the threshold .
Hence, show that any threshold low enough to carry processing over into day 2 is suboptimal.
Show carefully that the optimal price threshold is given by .
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1.II.29C
2008 comment(i) State the local existence theorem for the first order quasi-linear partial differential equation
which is to be solved for a real-valued function with data specified on a hypersurface . Include a definition of "non-characteristic" in your answer.
(ii) Consider the linear constant-coefficient case (that is, when all the functions are real constants and for some with real and real) and with the hypersurface taken to be the hyperplane . Explain carefully the relevance of the non-characteristic condition in obtaining a solution via the method of characteristics.
(iii) Solve the equation
with initial data prescribed on , for a real-valued function . Describe the domain on which your solution is and comment on this in relation to the theorem stated in (i).
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2.II.30C
2008 comment(i) Define the concept of "fundamental solution" of a linear constant-coefficient partial differential operator and write down the fundamental solution for the operator on .
(ii) State and prove the mean value property for harmonic functions on .
(iii) Let be a harmonic function which satisfies at every point in an open set . Show that if , then
Assume that . Deduce, by choosing and appropriately, that
[In (iii), is the ball of radius centred at
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3.II.29C
2008 commentLet be the space of smooth -periodic functions of one variable.
(i) For show that there exists a unique such that
(ii) Show that for every which is not identically zero, where is defined by
(iii) Show that the equation
with initial data has, for , a smooth solution such that for each fixed . Give a representation of this solution as a Fourier series in . Calculate and comment on your answer in relation to (i).
(iv) Show that for , and that as .
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4.II.30C
2008 comment(i) Define the Fourier transform of a Schwartz function , and also of a tempered distribution .
(ii) From your definition, compute the Fourier transform of the distribution given by
for every Schwartz function . Here is the integration element on the sphere of radius .
Hence deduce the formula of Kirchoff for the solution of the initial value problem for the wave equation in three space dimensions,
with initial data and , where . Explain briefly why the formula is also valid for arbitrary smooth .
(iii) Show that any solution of the initial value problem in (ii) is given by the formula derived in (ii) (uniqueness).
(iv) Show that any two solutions of the initial value problem for
with the same initial data as in (ii), also agree for any .
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1.II
2008 commentObtain an expression for the th term of an asymptotic expansion, valid as , for the integral
Estimate the value of for the term of least magnitude.
Obtain the first two terms of an asymptotic expansion, valid as , for the integral
[Hint:
[Stirling's formula may be quoted.]
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3.II
2008 commentDescribe how the leading-order approximation may be found by the method of stationary phase of
for , where and are real. You should consider the cases for which: (a) has one simple zero at , where ; (b) has more than one simple zero in the region ; and (c) has only a simple zero at .
What is the order of magnitude of if is non zero for ?
Use the method of stationary phase to find the leading-order approximation for to
[Hint:
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4.II
2008 commentThe Bessel equation of order is
Here, is taken to be an integer, with . The transformation converts (1) to the form
where
Find two linearly independent solutions of the form
where are constants, with , and and are to be determined. Find recurrence relationships for the .
Find the first two terms of two linearly independent Liouville-Green solutions of (2) for valid in a neighbourhood of . Relate these solutions to those of the form (3).
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1.II.31C
2008 commentDefine an integrable system in the context of Hamiltonian mechanics with a finite number of degrees of freedom and state the Arnold-Liouville theorem.
Consider a six-dimensional phase space with its canonical coordinates , , and the Hamiltonian
where and where is an arbitrary function. Show that both and are first integrals.
State the Jacobi identity and deduce that the Poisson bracket
is also a first integral. Construct a suitable expression out of to demonstrate that the system admits three first integrals in involution and thus satisfies the hypothesis of the Arnold-Liouville theorem.
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2.II.31C
2008 commentDescribe the inverse scattering transform for the equation, paying particular attention to the Lax representation and the evolution of the scattering data.
[Hint: you may find it helpful to consider the operator
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3.II.31C
2008 commentLet and be matrix-valued functions of depending on the auxiliary parameter . Consider a system of linear PDEs
where is a column vector whose components depend on . Derive the zero curvature representation as the compatibility conditions for this system.
Assume that
and show that (1) is compatible if the function satisfies the PDE
for some which should be determined.
Show that the transformation
forms a symmetry group of the PDE (2) and find the vector field generating this group.
Find the ODE characterising the group-invariant solutions of (2).
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1.II.32D
2008 comment(a) If and are operators which each commute with their commutator , show that . By considering
and differentiating with respect to the parameter , show also that
where .
(b) Consider a one-dimensional quantum system with position and momentum . Write down a formula for the operator corresponding to translation through , calculate , and show that your answer is consistent with the assumption that position eigenstates obey . Given this assumption, express the wavefunction for in terms of the wavefunction for .
Now suppose the one-dimensional system is a harmonic oscillator of mass and frequency . Show that
where are normalised wavefunctions with energies .
[Standard results for constructing normalised energy eigenstates in terms of annihilation and creation operators
may be quoted without proof.]
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2.II.32D
2008 commentDerive approximate expressions for the eigenvalues of a Hamiltonian , working to second order in the parameter and assuming the eigenstates and eigenvalues of are known and non-degenerate.
Let be angular momentum operators with joint eigenstates of and . What are the possible values of the labels and and what are the corresponding eigenvalues of the operators?
A particle with spin is trapped in space (its position and momentum can be ignored) but is subject to a magnetic field of the form , resulting in a Hamiltonian . Starting from the eigenstates and eigenvalues of this Hamiltonian when , use perturbation theory to compute the leading order corrections to the energies when is non-zero but much smaller than . Compare with the exact result.
[You may set and use ]
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3.II
2008 commentExplain, in a few lines, how the Pauli matrices with
are used to represent angular momentum operators with respect to basis states and corresponding to spin up and spin down along the 3 -axis. You should state clearly which properties of the matrices correspond to general features of angular momentum and which are specific to spin half.
Consider two spin-half particles labelled A and B, each with its spin operators and spin eigenstates. Find the matrix representation of
with respect to a basis of two-particle states . Show that the eigenvalues of the matrix are and find the eigenvectors.
What is the behaviour of each eigenvector under interchange of and ? If the particles are identical, and there are no other relevant degrees of freedom, which of the two-particle states are allowed?
By relating to the operator discussed above, show that your findings are consistent with standard results for addition of angular momentum.
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4.II.32D
2008 commentDefine the interaction picture for a quantum mechanical system with Schrödinger picture Hamiltonian and explain why either picture gives the same physical predictions. Derive an equation of motion for interaction picture states and use this to show that the probability of a transition from a state at time zero to a state at time is
correct to second order in , where the initial and final states are orthogonal eigenstates of with eigenvalues and .
Consider a perturbed harmonic oscillator:
with and annihilation and creation operators (all usual properties may be assumed). Working to order , find the probability for a transition from an initial state with to a final state with after time .
Suppose becomes large and perturbation theory still applies. Explain why the rate for each allowed transition is sharply peaked, as a function of , around .
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1.II.33E
2008 commentA beam of particles each of mass and energy scatters off an axisymmetric potential . In the first Born approximation the scattering amplitude is
where is the wave vector of the incident particles and is the wave vector of the outgoing particles at scattering angle (and ). Let and . Show that when the scattering potential is spherically symmetric the expression simplifies to
and find the relation between and .
Calculate this scattering amplitude for the potential where is a constant, and show that at high energies the particles emerge predominantly in a narrow cone around the forward beam direction. Estimate the angular width of the cone.
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2.II.33E
2008 commentConsider a large, essentially two-dimensional, rectangular sample of conductor of area , and containing electrons of charge . Suppose a magnetic field of strength is applied perpendicularly to the sample. Write down the Landau Hamiltonian for one of the electrons assuming that the electron interacts just with the magnetic field.
[You may ignore the interaction of the electron spin with the magnetic field.]
Find the allowed energy levels of the electron.
Find the total energy of the electrons at absolute zero temperature as a function of , assuming that is in the range
Comment on the values of the total energy when takes the values at the two ends of this range.
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3.II.33E
2008 commentConsider the body-centred cuboidal lattice with lattice points and , where and are positive and and take all possible integer values. Find the reciprocal lattice and describe its geometrical form. Calculate the volumes of the unit cells of the lattices and .
Find the reciprocal lattice vector associated with the lattice planes parallel to the plane containing the points and . Deduce the allowed Bragg scattering angles of X-rays off these planes, assuming that and that the X-rays have wavelength .
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4.II.33E
2008 commentExplain why the allowed energies of electrons in a three-dimensional crystal lie in energy bands. What quantum numbers can be used to classify the electron energy eigenstates?
Describe the effect on the energy level structure of adding a small density of impurity atoms randomly to the crystal.
-
2008
commentProve that energy fluctuations in a canonical distribution are given by
where is the absolute temperature, is the heat capacity at constant volume, and is Boltzmann's constant.
Prove the following relation in a similar manner:
Show that, for an ideal gas of monatomic molecules where , these equations can be reduced to
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3.II.34E
2008 commentDerive the following two relations:
and
[You may use any standard Maxwell relation without proving it.]
Experimentalists very seldom measure directly; they measure and use thermodynamics to extract . Use your results from the first part of this question to find a formula for in terms of the easily measured quantities
(the volume coefficient of expansion) and
(the isothermal compressibility).
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4.II.34D
2008 commentShow that the Fermi momentum of a gas of non-interacting electrons in volume is
Consider the electrons to be effectively massless, so that an electron of momentum has (relativistic) energy . Show that the mean energy per electron at zero temperature is .
When a constant external magnetic field of strength is applied to the electron gas, each electron gets an energy contribution depending on whether its spin is parallel or antiparallel to the field. Here is the magnitude of the magnetic moment of an electron. Calculate the total magnetic moment of the electron gas at zero temperature, assuming is much less than .
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1.II
2008 commentFrame is moving with uniform speed in the -direction relative to a laboratory frame . The components of the electric and magnetic fields and in the two frames are related by the Lorentz transformation
where and units are chosen so that . How do the components of the spatial vector (where ) transform?
Show that is obtained from by a rotation through about a spatial axis , where and should be determined. Hence, or otherwise, show that there are precisely two independent scalars associated with which are preserved by the Lorentz transformation, and obtain them.
[Hint: since there exists a unique real such that .]
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2008
commentThe retarded scalar potential produced by a charge distribution is given by
where denotes all 3 -space. Describe briefly and qualitatively the physics underlying this formula.
Write the integrand in the formula above as a 1-dimensional integral over a new time coordinate . Next consider a special source, a point charge moving along a trajectory so that
where denotes the 3 -dimensional delta function. By reversing the order of integration, or otherwise, obtain the Liénard-Wiechert potential
where and are to be determined.
Write down the corresponding formula for the vector potential .
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4.II
2008 commentThe Maxwell field tensor is given by
A general 4-velocity is written as , where , and . A general 4-current density is written as , where is the charge density and is the 3 -current density. Show that
In the rest frame of a conducting medium, Ohm's law states that where is the conductivity. Show that the relativistic generalization to a frame in which the medium moves with uniform velocity is
Show that this implies
Simplify this formula, given that the charge density vanishes in the rest frame of the medium
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1.II.35E
2008 commentFor the metric
obtain the geodesic equations of motion. For a massive particle show that
for some constant . Show that the particle moves on trajectories
where is the proper time, if the origins of are chosen appropriately.
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2.II.35E
2008 commentLet be a path with tangent vector . For vectors and defined on let
where is the metric connection for a metric is defined similarly. Suppose is geodesic and is an affine parameter. Explain why . Show that if then is constant along .
If is a family of geodesics which depend on , let and define
Show that and obtain
What is the physical relevance of this equation in general relativity? Describe briefly how this is relevant for an observer moving under gravity.
[You may assume .]
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4.II.36E
2008 commentA solution of the Einstein equations is given by the metric
For an incoming light ray, with constant , show that
for some fixed and find a similar solution for an outgoing light ray. For the outgoing case, assuming , show that in the far past and in the far future .
Obtain the transformed metric after the change of variables . With coordinates sketch, for fixed , the trajectories followed by light rays. What is the significance of the line ?
Show that, whatever path an observer with initial takes, he must reach in a finite proper time.
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1.II.36A
2008 commentDerive the relation between the stress tensor and the rate-of-strain tensor in an incompressible Newtonian fluid, using the result that there is a linear dependence between the components of and those of that is the same in all frames. Write down the boundary conditions that hold at an interface between two viscous fluids.
Viscous fluid is contained in a channel between the rigid planes and . The fluid in has dynamic viscosity , while that in has dynamic viscosity . Gravity may be neglected. The fluids move through the channel in the -direction under the influence of a pressure gradient applied at the ends of the channel. It may be assumed that the velocity has no -components, and all quantities are independent of .
Find a steady solution of the Navier-Stokes equation in which the interface between the two fluids remains at , the fluid velocity is everywhere independent of , and the pressure gradient is uniform. Use it to calculate the following:
(a) the viscous tangential stress at and at ; and
(b) the ratio of the volume fluxes of the two different fluids.
Comment on the limits of each of the results in (a) and (b) as , and as
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2.II.36A
2008 commentViscous fluid with dynamic viscosity flows with velocity (in cartesian coordinates in a shallow container with a free surface at . The base of the container is rigid, and is at . A horizontal stress is applied at the free surface. Gravity may be neglected.
Using lubrication theory (conditions for the validity of which should be clearly stated), show that the horizontal volume flux satisfies the equations
where is the pressure. Find also an expression for the surface velocity in terms of and .
Now suppose that the container is cylindrical with boundary at , where , and that the surface stress is uniform and in the -direction, so with constant. It can be assumed that the correct boundary condition to apply at is , where is the unit normal.
Write , and show that satisfies the equation
Deduce that if (constant) then . Find in this case.
Now suppose that , where . Verify that to leading order in for some constant to be determined. Hence determine up to and including terms of order .
[Hint: for any vector field
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3.II.36A
2008 commentShow that, in cylindrical polar co-ordinates, the streamfunction for the velocity and vorticity of two-dimensional Stokes flow of incompressible fluid satisfies the equations
Show also that the pressure satisfies .
A stationary rigid circular cylinder of radius occupies the region . The flow around the cylinder tends at large distances to a simple shear flow, with velocity given in cartesian coordinates by . Inertial forces may be neglected.
By solving the equation for in cylindrical polars, determine the flow field everywhere. Determine the torque on the cylinder per unit length in .
[Hint: in cylindrical polars
The off-diagonal component of the rate-of-strain tensor is given by
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4.II.37A
2008 commentViscous incompressible fluid of uniform density is extruded axisymmetrically from a thin circular slit of small radius centred at the origin and lying in the plane in cylindrical polar coordinates . There is no external radial pressure gradient. It is assumed that the fluid forms a thin boundary layer, close to and symmetric about the plane . The layer has thickness . The -component of the steady Navier-Stokes equations may be approximated by
(i) Prove that the quantity (proportional to the flux of radial momentum)
is independent of .
(ii) Show, by balancing terms in the momentum equation and assuming constancy of , that there is a similarity solution of the form
where are constants. Show that for suitable choices of and the equation for takes the form
(iii) Give an inequality connecting and that ensures that the boundary layer approximation is valid. Solve the equation to give a complete solution to the problem for when this inequality holds.
[Hint:
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1.II
2008 commentShow that in an acoustic plane wave the velocity and perturbation pressure are everywhere proportional and find the constant of proportionality.
Gas occupies a tube lying parallel to the -axis. In the regions and the gas has uniform density and sound speed . For the gas is cooled so that it has uniform density and sound speed . A harmonic plane wave with frequency is incident from . Show that the amplitude of the wave transmitted into relative to that of the incident wave is
where and .
What are the implications of this result if ?
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2.II.37B
2008 commentShow that, in one-dimensional flow of a perfect gas at constant entropy, the Riemann invariants are constant along characteristics .
A perfect gas occupies a tube that lies parallel to the -axis. The gas is initially at rest and is in . For times a piston is pulled out of the gas so that its position at time is
where is a constant. Sketch the characteristics of the resulting motion in the plane and explain why no shock forms in the gas.
Calculate the pressure exerted by the gas on the piston for times , and show that at a finite time a vacuum forms. What is the speed of the piston at ?
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3.II
2008 commentThe real function satisfies the Klein-Gordon equation
Find the dispersion relation for disturbances of wavenumber and deduce their phase and group velocities.
Suppose that at
Use Fourier transforms to find an integral expression for when .
Use the method of stationary phase to find for for fixed . What can be said if ?
[Hint: you may assume that
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4.II B
2008 commentA layer of rock of shear modulus and shear wave speed occupies the region with a free surface at . A second rock having shear modulus and shear wave speed occupies . Show that elastic waves of wavenumber and phase speed can propagate in the layer with zero disturbance at if and satisfies the dispersion relation
Show graphically, or otherwise, that this equation has at least one real solution for any value of , and determine the smallest value of for which the equation has at least two real solutions.
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1.II.38C
2008 commentThe Poisson equation in the unit square , with zero boundary conditions on , is discretized with the nine-point formula
where , and are grid points.
(a) Prove that, for any ordering of the grid points, the method can be written as with a symmetric positive-definite matrix .
(b) Describe the Jacobi method for solving a linear system of equations, and prove that it converges for the above system.
[You may quote without proof the corollary of the Householder-John theorem regarding convergence of the Jacobi method.]
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2.II.38C
2008 commentThe advection equation
is solved by the leapfrog scheme
where and is the Courant number.
(a) Determine the local error of the method.
(b) Applying the Fourier technique, find the range of for which the method is stable.
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3.II.38C
2008 comment(a) A numerical method for solving the ordinary differential equation
generates for every a sequence , where is an approximation to and . Explain what is meant by the convergence of the method.
(b) Prove from first principles that if the function is sufficiently smooth and satisfies the Lipschitz condition
for some , then the trapezoidal rule
converges.
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4.II.39C
2008 commentLet be a real matrix with linearly independent eigenvectors. When calculating eigenvalues of , the sequence , is generated by power method , where is a real nonzero vector.
(a) Describe the asymptotic properties of the sequence , both in the case where the eigenvalues of satisfy , and in the case where . In the latter case explain how the (possibly complexvalued) eigenvalues and their corresponding eigenvectors can be determined.
(b) Let , and suppose that, for a large , we obtain the vectors
Find two eigenvalues of and their corresponding eigenvectors.